2 : . .9 1 lillill~lllllln 92-01188 i|1111 · pdf filesiae 1204, aifrix va 2212-4 ... false...
TRANSCRIPT
LOAN DOCUMENT _____
PHOTOGRAPH THIS SHEET
Lfl:5LEVL INVENTORY
z
00
DOCUMENT IDENTIFICATION
A
~D
DISTRIUTION STATEMENT L
D11C TAC 0" D TIC_LNAVNOUNCED 0 ELECTD
MAR 3 1992W
I)ISTRIOl TION! C TAVAIL.ABILITY CO0DEcS
I nSI .BTON AVALAiILITY AND, I . VlF .
DATE ACCESSIONED
_ __\_ c
ADISTRIBUTION STAMP
RE
DATE RETURNED
92-01188g 2 1. : . ) .9 1 Lillill~llllllN I|1111 liIIDATE RECEIVED IN DTIC REGISTERED OR CERTIFIED NUMBFR
PHOTOGRAPH THIS SHEET AND RETURN TO DTIC-FDAC
)TIC,"" 70A DOCUMFIRr PROCESSING SHELr PREV EDITOS IUS MAY B USDH) U-NTII
9TrOC 1 EX1HAUSrD
LOAN DOCUMENT
AD-A247 350
RL-TR-91 -80in-House ReportJune 1991
IMPROVEMENT IN RADAR DETECTIONUNDER SCAN TO SCAN PROCESSING ANDSCAN RATE REDUCTION
Dr. Donald D. Weiner, RI/OCTS
Dr. Joon Song, Syracuse University
APPROVED FOR PUB/C RELASE- /S TRIBUTYN IULIME.
Rome LaboratoryAir Force systems command
Griffiss Air Force Base, NY 13441-5700
This report has been reviewed by the Rome Laboratory Public Affairs
Office (PA) and is releasable to the National Technical Information
Service (NTIS). At NTIS it will be releasable to the general public,
including foreign nations.
RL-TR-91-80 has been reviewed and is approved for publication.
APPROVED:
FRED J. DEMMAChief, Surveillance Technology DivisionDirectorate of Surveillance
APPROVED:
JAMES W. YOUNGBERG, Lt Col, USAFDeputy Director of Surveillance
FOR THE COMMANDER:
JAMES W. HYDE III
Directorate of Plans & Programs
If your address has changed or if you wish to be removed from the Rome
Laboratory mailing list, or if the addressee is no longer employed by
your organization, please notify RL (OCTS ) Griffiss AFB, NY 13441-5700.
This will assist us in maintaining a current mailing list.
Do not return copies of this report unless contractual obligations or
notices on a specific document require that it be returned.
Form ApproveaREPORT DOCUMENTATION PAGE 0MB No. 0704-0188P,±10 rep rev ig ipan for &is - c- &v -i *wo s etrmed to aveage I -ou pe respon rcd ng rt , n'f re , j . sea'-Ig existrg c3a soucesgJwrga rirwtw" t* da rd.eeded an wVz rg and revmwrg c- ;iemor c irnmor Sc2 Trrtrws regacrdng o t. nesfrnaS orany other asect '#,-coij rar , ci grcyYg r SJgr g e fcr recd"cr V ti boe' w to Waru an Haamits Savva Dreotoa for r soNTrr, Operajs andReports. 1215 .JeffersonDaws HkIwa. SiAe 1204, Aifrix VA 2212-4 ar cot" Oflca d M sgjwrt ano Buget Papwo Rmdjcwn Pro (0704-0 881, Wamrgt\ DC 2C5Ca
1. AGENCY USE ONLY (Leave Blank) 2 REPORT DATE 3 REPORT TYPE AND DATES COVEREDJune 1991 In-House Jan 90 - Dec 90
4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
IMPROVEMENT IN RADAR DETECTION UNDER SCAN TO SCAN PE 62702FPROCESSING AND SCAN RATE REDUCTION PE - 4506
6. AUTHOR(S) TA PROJDr. Donald D. Weiner - Rome Laboratory (OCTS) *Dr. Joon Song - Syracuse University
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Rome Laboratory (OCTS) REPORT NUMBER
Griffiss AFB NY 13441-5700 RL-TR-91-80
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORINGIMONITORING
AFOSR/XOT AGENCY REPORT NUMBER
Boiling AFB DC 20332-6448 N/A
11. SUPPLEMENTARY NOTES (see reverse)
Rome Laboratory Project Engineer: Michael C. Wicks/OCTS/(315) 330-4437*This PhD dissertation research was performed by Dr. Joon Song of Syracuse
1 2a DISTRIBUTiON/AVAJLABIUTY STATEMENT 1 2b. DISTRIBUTION CODE
Approved for public release; distribution unlimited.
13. ABSTRACT (Mm.ri2m -arc&)
Recent advances in signal processing technology have resulted in a significantlygreater computational capability at lower cost. As a result, many investigations arecurrently under way to determine how increased signal processing can be used to improvethe detection performance of existing radars without requiring major equipmentmodifications. Two approaches towards this goal are examined: The first is a scan-to-scan processing (SSP) also known as track-before-detect (TBD). The second involvesscan rate reduction along with efficient post detection integration. Detailedanalyses are carried out for the two methods.
4. SUBJECT TERMS 1S f,-MBER or PAGES
328Signal Processing, radar, tracking & PRICE CODE
7. SECURITY CLASSIFICATION 18. SECURITY Ct NSIFICA]I ON I C .r C1_; S S 1!7! ^ 11, ^ T !N1oU IITO 1A SRCE qtT .... ... ?2, UMIFATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT
UNCLASSIFIED UNCLASSI7 IED UNCLASSIFIED ULNSN 754001 2S-5M Stancar Form 298 (Or. 2 er
Pres rided by ANSI Sta Z2g 829102
Block 11 (Cont'd)
and his advisor, Dr. Donald D. Weiner of Rome Laboratory. Dr. Weiner is a
visiting scientist at the Rome Laboratory under AFOSR sponsorship.
March 7, 1991
Rome Laboratory/OCTS (M. Wicks)FAX No. (315) 336-7352
I hereby grant permission to Rome Laboratory to publish mydissertation as a government technical report coauthored byDr. D. Weiner and myself.
I also agree to the copyright clause, "This material maybe reproduced by or for the US government."
S cerely,
6551 24th Ave. N.E.Seattle, WA 98115
CONTENTS
Page
LIST OF TABLES v.
LIST OF FIGURES ........................................ vii.
CHAPTER
1.0 INTRODUCTION ........................................ 1
Problem Statement ................................ 1
Description of the Airborne Long Range Surveil-lance Radar Under Consideration ................ 3
System Constraints ...................... 8
Performance Improvement Criteria ........ 20
Dissertation Outline ... ............. 27
2.0 SOME CONSIDERATIONS IN PERFORMANCE COMPARISON OF
MODIFIED AND BASELINE CONFIGURATIONS .............. 29
Definition of Terms ............................. 29
Basic Assumptions ......................... 40
Number of CPI's in a Beam Dwell and Number ofAvailable Pulses in a CPI ....................... 42
Beam Shape Loss ....... .. ..... ................ 47
False Alarm Probability Allocation and theSystem False Alarm Verification ................. 55
Target Models Assumed ........................... 79
3.0 PERFORMANCE OF THE AIRBORNE SURVEILLANCE RADAR
IN ITS BASELINE CONFIGURATION ...................... 86
Sufficient Statistic and the Likelihood Ratio...105
Detection Threshold and Probabilities of Detect-ion ............................................ 105
Detection Performance of the Baseline Config-uration ........................................ 115
iii
Effect of Beam Shape Loss on Detection Proba-
bility ......................................... 129
4.0 SYSTEM PERFORMANCE WITH SCAN-TO-SCAN PROCESSING ..140
Description of Scan-to-Scan Processing ......... 141
Scan-to-Scan Correlation Window ................ 149
Determination of Threshold Setting and Depth ofScan from False Alarm Considerations ........... 158
Detection Performance under Modified J of KSSP ............................................ 170
Detection Performance under ConventionalJ of K SSP ..................................... 197
5.0 PERFORMANCE IMPROVEMENT WITH SCAN RATE REDUCTION
AND NONCOHERENT INTEGRATION ...................... 207
Scan Rate Optimization ......................... 211
Scan Rate Optimization for a High PRF Radar .... 227
Detection Probability with NoncoherentIntegration in Lieu of Binary Integration ...... 239
Probabilities of Detection with Slower ScanRate ........................................... 244
6.0 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK ...... 265
Summary ............................. ........... 265
Recommendation for Future Work ................. 268
Clutter Model ...................... 274
Clutter Cancellation Techniques .............. 288
BIBLIOGRAPHY ............................................ 304
iv
LIST OF TABLES
Table Page
2.3-1. Number of Pulses Available for Integration with
a 6 rpm Antenna Scan Rate ........................ 46
2.4-1. Summary of Beam Shape Loss for the 2-SLant and
3-Slant Configurations ........................... 54
2.5-1. False Alarm Probability for the Processing Opt-
ions Evaluated .............................. 78
3.4-1. Detection Probabilities with 2 of 2, 2 of 3, and
2 of 4 Processing ............................... 139
4.2-1. SSP correlation window and False Alarm Oppor-
tunity Multiplier ............................... 156
4.4-1. Ways to Fail in J=2 of K<L=6 SSP ................ 178
4.4-2. Ways to Fail in J=3 of K<L=6 SSP ................ 179
4.4-3. An Illustration for Computing Pcjw/ SSP by Two
Formulas (3-Slant Configuration with 6-Scan
Deep SSP) ....................................... 181
4.4-4. Expressions for Cumulative Detection Probability
with Scan-to-Scan Processing .................... 183
4.4-5. Cumulative Probability of Detection vs. SNR,
3-Slant, Normal Time Overhead, as a Function of
Scan Depth Used ................................. 184
4.4-6. Probability of Detection vs. SNR, 3-Slant, Normal
Time Overhead .................................... 187
4.4-7. Probability of Detection vs. SNR, 3-Slant,
Reduced Time Overhead, Swerling Case 1 .......... 196
4.5-1. Single Scan Probability of Detection vs. SNR,
2-Slant with a Reduced Time Overhead ............ 199
V
Table Page
4.5-2. Cumulative Probability of Detection vs. SNR,
2-Slant with a Reduced Time Overhead ............ 201
5.2-1. Optimum Values A/R 1 and Corresponding Values of
RM/R1 - 3 of 3 Post Detection Integration ....... 236
5.2-2. Optimum Values A/R 1 and Corresponding Values of
Rm/R1 by Mallett and Brennan .................... 237
5.4-1. Single Scan Detection Probability @ 3 rpm Antenna
Scan Rate with Normal Time Overhead ............. 250
5.4-2. Required Pd to Achieve Pc= 0 .9 in L Scans ........ 255
5.4-3. Summary of Performance Improvement with the Ant-
enna Scan Rate of 3 rpm for a Swerling Case 1
Target .............. ............................ 257
6.2-1. Radar Characteristics and Clutter Sample Size ...275
6.2-2. Range Cell to Range Cell Variation of Clutter
Statistics ...................................... 287
vi
LIST OF FIGURES
Figure Page
1.1-1. Examples of Airborne Surveillance Radar System .... 2
1.1-2. Simplified Block Diagram for Radar Receiver/
signal Processor ............................. 4
1.1-3. Surveillance Volume for an Airborne Radar ........ 22
1.1-4. Detection Scenario of an Approaching Target with
a Constant Radial Velocity ....................... 25
2.1-1. Modulation Period ........................... 33
2.1-2. Received Pulse Returns in a High PRF Airborne
Radar ............................................ 34
2.1-3. A Comparison ot IIR and FIR Clutter Canceller
in a High PRF Waveform .......................... 36
2.4-1. Possible Positions of a Set of modulation Periods
w.r.t. Beam Centerline ........................... 51
2.5-1. Process by which False Alarms Occur .............. 65
3.1-1. Assumed ADC and Clutter Canceller Character-
istics ........................................... 86
3.1-2. Equivalent Receiver Block Diagram ................ 89
3.2-1. Processes for Determining Probabilities of
Detection after M of N Post Detection Binary
Integration ..................................... 108
3.2-2. Processes for Determining Probabilities of
Detection after M of N Post Detection Binary
Integration (Slant-to-Slant RCS Fluctuation) .... 108
3.3-1. Detection Performance of the Baseline, Swerling
and Marcum Target Models ........................ 116
vii
Figure rage
3.3-2. Cumulative Detection Probabilities for Swerling
and Marcum Target Models ........................ 118
3.3-3. Detection Performance of the Baseline, Swerli~g
Case 1 Target Model .............................. 119
3.3-4. Detection Performance of the Baseline, Swerling
Case 2 Target Model .............................. 120
3.3-5. Detection Performance of the Baseline, Swerling
Case 3 Target Model ............................. 121
3.3-6. Detection Performance of the Baseline, Swerling
Case 4 Target Model ........................... .122
3.3-7. Detection Performance of the Baseline, Non-
Fluctuating (Marcum) Target Model ................ 123
3.3-8. SNR vs. Probability of Detection, 3 of 3 Pro-
cessing @ 3 rpm, Normal Time Overhead ........... 125
3.3-9. SNR vs. Probability of Detection, 3 of 3 Pro-
cessing @ 3 rpm, Normal Time Overhead, with
Average Beam Shape Loss ......................... 126
3.3-10. SNR vs. Probability of Detection, 3 of 3 Pro-
cessing @ 3 rpm, Reduced Time Overhead .......... 128
3.4-1. Beam Shape loss for 2 of 2, 2 of 3, and 2 of 4
Processing given a 3-Slant in a Beamwidth ....... 134
3.4-2. Matrix of Possible Combination of 2 of 4 Detect-
ions on a 3-Slant Sliding Window ................ 137
4.2-1. Correlation Window for the ith Scan ............ 146
4.2-2. Maneuver Scenario used for Testing the SSP
Correlation Window .............................. 155
viii
Figure Page
4.3-1. Flight Test Verification of the Prediction for
False Alarms before SSP with 2 of 3 Post Detection
Integration ...................................... 162
4.4-1. SNR vs. Probability of Detection, 3-Slant
System, Swerling Case 1 Target .................. 191
4.4-2. SNR vs. Probability of Detection, 3-Slant
System, Swerling Case 2 Target .................. 192
4.4-3. SNR vs. Probability of Detection, 3-Slant
System, Swerling Case 3 Target .................. 193
4.4-4. SNR vs. Probability of Detection, 3-Slant
System, Swerling Case 4 Target .................. 194
4.4-5. SNR vs. Probability of Detection, 3-Slant
System, Nonfluctuating (Marcum) Target .......... 195
4.5-1. SNR vs. Single Scan Probability of Detection,
2-Slant System, Reduced Time Overhead ........... 200
4.5-2. SNR vs. Cumulative Probability of Detection (2 of
6 SSP), 2-Slant System, Reduced Time Overhead ...202
4.5-3. SNR vs. Cumulative Probability of Detection (3 of
6 SSP), 2-Slant System, Reduced Time Overhead ...203
4.5-4. Illustration of Performance Comparison Method ...204
5.1-1. Detection Scenario of an Approaching Target ..... 212
5.1-2. Single Scan Probability of Detection, vs. SNR,
1.2 rpm Antenna Scan Rate ....................... 217
5.1-3. Cumulative Probability of Detection vs. SNR,
1.2 rpm Antenna Scan Rate ....................... 217
5.4-1. SNR vs. Probability of Detection, 3-NCI @ 3 rpm,
Normal Time Overhead ............................ 246
ix
Figure Page
5.4-2. SNR vs. Probability of Detection, 6-NCI @ 3 rpm,
Normal Time Overhead ............................ 247
5.4-3. SNR vs. Probability of Detection, 3 of 3 Pro-
cessing, @ 3 rpm, Normal Time Overhead .......... 248
5.4-4. SNR vs. Probability of Detection, 3 of 6 Pro-
cessing, @ 3 rpm, Normal Time Overhead .......... 249
5.4-5. Cumulative Detection Probability in 20 seconds:
3-NCI @ 3 rpm vs. 3 of 3 @ 6 rpm ................ 261
5.4-6. Cumulative Detection Probability in 60 seconds:
3-NCI @ 3 rpm vs. 3 of 3 @ 6 rpm ................ 262
5.4-7. Average Number of Track Updates in 5 Minutes,
3-NCI @ 3 rpm vs. 3 of 3 @ 6 rpm ................ 264
6.2-1. Histograms of the Normalized mean Values of
Clutter Returns ................................. 280
6.2-2. Scaled Gamma Density Function with Shape Factor
of 1 and 2 ...................................... 281
6.2-3. Histograms of Normalized Clutter Returns for
Individual Range Cells .......................... 283
6.2-4. Averaged Histogram of Normalized Clutter
Returns of a Range Cell ......................... 285
6.2-5. Candidate Rician and Gamma density for the
Conditional Density for the clutter ............. 286
6.2-6. Performance Comparison of Clutter Cancellers:
IIR vs. FIR with Binomial Coefficients as
Weights ......................................... 292
6.2-7. Performance of Delayline Cancellers with Binomial
Coefficients as Weights: 2, 3, and 4 Stages ..... 294
x
Figure Page
6.2-8. Performance of Delayline Cancelers with Binomial
Coefficients as Weights: 2 vs. 7 stage .......... 296
6.2-9. Performance Comparisons of Clutter Canceller:
IIR vs. FIR with Eigen Vector as Weights ........ 297
6.2-10. Gain vs. Frequency of the FIR Filter with
Eigen Vector as Weights ......................... 302
xl
CHAPTER 1
INTRODUCTION
1.1 Problem Definition
It takes enormous technical and financial resources and commitment to successfully
develop a long range surveillance radar. Two successful examples are the US Navy E-2C
Hawkeye and US Air Force E-3B AWACS Surveillance Radars. These airborne
surveillance radar systems are pictured in Figure 1.1-1. It is said that the success rate of a
major radar development program from conception to deployment is on the order of 1%
[1]. Therefore, when a new threat emerges, it is preferable with respect to cost and risk to
seek improvements in detection performance of an existing and proven radar system rather
than embarking on a totally new radar development program.
This report presents for the first time a detailed analysis of the Scan-to-Scan
Processing (SSP) concept which has been pursued over the past ten or more years through
several independent industrial research and development programs as a means for
improving radar detection performance.
The past efforts to evaluate performance have primarily relied on the manipulation
of raw radar data at the output of the analog-to-digital converter with a computer simulation
package. Although significant improvement has been claimed, the careful analysis and
interpretation of the results presented in this, report reveals that only a marginal
improvement can be attributed to the Scan-to-Scan Processing. Consequently, an alternate
approach which relies on scan rate reduction and noncoherent integration (NCI), in lieu of
the M of N binary post detection integration widely used today, is proposed and
investigated. It is shown that slowing down the scan rate coupled with noncoherent
integration, can potentially deliver higher performance improvement with less risk.
m =.m., ,,, lra nl m IIII~llli1
fC.
(a) USAF E-3 AWACS AEW Aircraft with Rotodome Antenna
(b) USN E-2C Hawkeye AEW Aircraft with Rotodorne Antenna
Figure 1.1-1
Examples of Airborne Surveillance Radar Systems
2
In this section a description of the radar under consideration in its baseline
configuration is presented first. The constraints under which improvement is sought are
then specified. Finally, the criterion by which the improvement is measured is discussed.
1.1.1 Description of the Airborne Long Range Surveillance Radar under Consideration
A brief description of the receiver/processor portion of the radar system under
consideration in its baseline configuration is given here. Performance of this receiver/
processor will provide a basis for determining any performance improvement in the
modified configuration. The radar for which a detection improvement is sought is a high
PRF, pulse doppler radar with a superheterodyne receiver followed by an analog-to-digital
converter (ADC) and digital signal processor section. A basic receiver and processor block
diagram, common to all pulse doppler radars, is shown in Figure 1.1-2.
A superheterodyne receiver typically has two frequency down conversion stages to
reach the final intermediate frequency (IF). Not only is amplification at IF less costly and
more stable than at a microwave frequency, but the wider percentage bandwidth occupied
by the desired signal simplifies the filtering operation. In addition, the superheterodyne
receiver allows variation of the local oscillator frequency to follow any desired or
unintentional tuning variation of the transmitter without disturbing the filtering at IF. The
reference signal applied to the first mixer is provided by the STALO (stable local oscillator)
while the reference signal applied to the second mixer is provided by the COHO (coherent
local oscillator). The STALO signal has the frequency stability required for coherent
processing while the COHO can be used to introduce phase correction needed to
compensate for radar platform motion or transmitter phase variations.
A description of the flow of the received signal (target signal plus clutter plus other
interference such as jamming noise when present) is presented below. A detailed analysis
3
r 1
0 ~
1433
4J
P- 0
u ) Un
tn 0
x P t r4,0 LLL.
E-EC-(%
CU 41,- r4 0
>~ 41
00
4134
r- - <
Lft 4-
01 4-
t-) E U
a c4
leading to probabilities of detection for Swerling and Marcum target models is presented in
Chapter 3.
It is assumed that the carrier frequency of the received signal is (fo+fd+fmbc)
where fo is the carrier frequency of the transmitted signal, fd is the doppler shift of a
moving target as it would be observed from a motionless platform, and fmbc is the
doppler shift of the return from a ground patch in the direction of the main beam due to the
platform motion along the line of antenna pointing direction. The thermal noise generated
from all sources is modeled by the additive white Gaussian noise process, nl(t). In
practice, the low noise amplifier (LNA) at the receiver front end essentially determines the
noise level. The low pass filter following the first mixer separates out the signal at the
difference frequency. The frequency offset fmbc can be incorporated into the COHO signal
so as to center the mainbeam clutter spectrum about the IF frequency, fIF.
The bandwidth of the band-pass filter at fIF is equal to 2B where B is the half
power signal bandwidth. For analytic convenience, this filter is assumed to be an ideal
band-pass filter whose gain is zero outside the pass band. This filter has no effect on the
band-limited input signal but limits the bandwidth of the noise process. It forms a part of
the matched filter which also includes the two paths for the quadrature components. The
down-converter, sometimes referred to as a synchronous detector, converts the signal at IF
to baseband while preserving both phase and amplitude information.
The analog-to-digital converter (ADC) provides a means for signal processing in
the digital domain with the attendant advantage in flexibility, reliability, repeatability and
precision. Its bandwidth, linearity, and dynamic range are important factors for coherent
signal processing and is often the limiting factor in the bandwidth and instantaneous
dynamic range of the system. In a pulse doppler radar, particularly with a high PRF
5
waveform, the mainbeam clutter usually sets the system dynamic range requirements. The
clutter canceller is another critical signal processing element in a pulse doppler radar. If
not cancelled, even with an infinite dynamic range processor downstream, the mainbeam
clutter will appear as residue clutter in the otherwise clutter free doppler zone as a result of
integration through doppler filter sidelobes.
The clutter canceller in this radar is a cascade of two second order recursive digital
filter sections, each with two poles and two zeros. This is designed to provide a desired
shaped velocity response in the frequency domain. The drawback is that a significantly
long time domain transient response is generated during which signal integration cannot be
performed. Other forms of clutter cancellers are multistage delay line cancellers with
binomial coefficients as weights, and the optimum linear filter whose weights are given by
the elements of the eigen vector corresponding to the minimum eigen value of the clutter
correlation matrix. Some analysis and recommendations for further work on these filters
are given in chapter 6.
When a coded waveform having a long time-bandwidth product is transmitted so as
to increase energy without exceeding the transmitter peak power, a pulse compressor will
follow the clutter canceller. This restores the range resolution capability of the radar by
generating an equivalent narrow pulse. In effect, the pulse compressor is a matched filter
for the coded waveform. The doppler filter bank is implemented by a two rail pipeline FF1
(fast Fourier transform) and performs a 128 point discrete Fourier transform where the
input samples are windowed with a 42 dB Hamming weight. Not shown in the block
diagram is the corner turning memory before the FFT which selects all of the pulses for a
given range gate from the sequence of transmitted pulse returns where all of the range
information is contained. The receiver/processor up to this point is assumed to be linear.
6
For each range gate and each doppler filter cell the inphase channel output and the
quadrature channel output multiplied by -j are added together, and the real and the
imaginary parts are separated, squared and summed, and then square rooted to obtain the
envelope voltage. This becomes a single observation variable in the decision space. That
is, all the pulses which are passed into the coherent integrator (FF1) result in a single pulse
for each range-doppler cell in the radar detection context. Also, this is the sufficient
statistic resulting from the Bayes likelihood ratio test for well known Swerling and
Marcum target models with unknown initial phase embedded in white Gaussian noise.
For these signal plus noise statistics, the above described receiver is the optimum receiver
that maximizes the probability of detection. Its derivation is given by DiFranco and Rubin
[2]. The test is performed separately for every azimuth-range-doppler cell.
The constant false alarm rate (CFAR) circuit provides the noise average to be used
to determine the test threshold. False alarm rate is very sensitive to threshold level. For
example, one dB change in the threshold can result in three orders of magnitude change in
false alarm probability. To prevent fluctuations in the false alarm rate, actual noise
averages are used to determine the threshold. Typically a cell averaging CFAR is used
whereby envelope voltages of a predetermined number of range cells on both sides of the
range cell under test are used for averaging. The two cells immediately adjacent to the test
cell are excluded from the CFAR block. This average is multiplied by a fixed constant and
is used as a threshold to decide whether the test statistic is signal plus noise (when above
the threshold) or noise alone (when below the threshold) in the target detector.
In this radar, received radar pulses in a beam dwell are divided into three coherent
processing intervals (CPI's) each operating at a different pulse repetition frequency (PRF)
and each of which results in an observation variable for every range-doppler cell for
detection decision making. The different PRFs are used so that range ambiguities can be
7
resolved by means of the Chinese remainder theorem. Before the post detection binary
integration, filter normalization is performed to eliminate the variability in doppler
frequency for a specific doppler cell. The M of N post detection integrator used in this
radar requires that all three of the observation samples corresponding to a particular
doppler filter in a beam dwell must have exceeded the threshold before the triplet can be
declared as a candidate target with the velocity corresponding to that doppler filter.
Target report processing includes filter unfolding to resolve whether the velocity is
opening or closing, and range ambiguity resolution to determine the unambiguous range for
each candidate target. Azimuth, range and filter centroiding, deghosting to remove false
target reports when more than one detection occurs in a doppler filter, and coordinate
conversion to a stabilized coordinate system are part of the target report processing
function. The results are sent out as radar reports which can be displayed on an operator
console screen directly or after further processing by the Kalman filter tracker.
1.1.2 System Constraints
The system constraints to be observed in seeking detection performance
improvement are specified in this section. Specifically, the antenna and transmitter are not
allowed to be changed in the proposed modifications. As a result, the following system
parameters are held fixed:
1. Antenna gain and radiation pattern
2. Transmitter power and duty factor
3. Radar operating frequency (carrier frequency and its operating frequency range).
The reason for these constraints is obvious. Changes to any of the above require a
major new development with the associated cost, schedule and technical risks. Under these
constraints, the potential area for investigation that would yield detection performance
8
improvement lies in signal processing changes whose implementation has been greatly
facilitated by the recent breakthroughs in signal processing hardware and memory
devices. As a consequence, the Scan-to-Scan Processing (SSP) concept, similar to Track
Before Detect (TBD), has been promoted as a promising concept for significant detection
performance improvement. However, a detailed analysis to support the projected benefits
and associated drawbacks has not been carried out. A rigorous analysis of the SSP concept
is one of the accomplishments of this dissertation. But first, the impact of the power-
aperture product constraint is reviewed so as to provide awareness on the limits of detection
performance improvement possible when the fundamental radar asset, namely the power-
aperture product, is fixed.
A 'rule of thumb' in radar engineering is that specification of power-aperture
product, surveillance volume and the search frame time determines the detection
performance against a given target, regardless of operating frequency, ignoring the
secondary effects such as frequency dependent system losses, etc. The search frame time
is the time it takes for the antenna beam to sweep the specified surveillance volume once at
a predetermined rate. That the limit of detection performance is set by the above
parameters can be seen by applying the radar range equation to the specified surveillance
volume to be searched and utilizing the relationship between the solid angle subtended by
the antenna beam and its aperture.
To see this, a simple radar range equation is first developed for the case where the
dwell time or the number of pulses available for integration is given. If the transmitter
power Pt is radiated isotropically, the power density produced at range R is given by
power per unit) Ptarea at range R = 4-R2
9
If a transmitting antenna of one way power gain Gt is used, the power density in
the direction of this gain is
power per unit area at range = Pt GtR in the given direction j 41R 2
The signal strength reflected toward the radar per unit solid angle from a target of radar
cross section (RCS) a illuminated by the transmitted wave at range R is
reflected power per unit solid angle_ PtG t x _ain the direction of the radar 4nR2 4n
The reflected wave arrives back at the radar with a power density given by
unit area at the receiver- 4xR 2 49 R2
The amount of power intercepted by the receiving antenna is this power density multiplied
by the effective aperture defined by
effective 2=A GX 2
receiving aperture4 (1.1-1)
where X is the radar wave lei.gth. The received signal power, S, is then expressed as
s , x2S = PP3 GrX 2a
(4t)3 R4 (1.1-2)
10
It is convenient to introduce a concept of equivalent input noise level, N, to arrive
at the output signal-to-noise ratio at the end of the linear portion of the receiver before
envelope detection. To facilitate this, the noise factor, NF, and effective noise temperature,
Te, of an amplifier or a receiver with a gain G are introduced. The noise factor is defined
as
NF = SN) = No,(S/N)out kToBG
where k is the Boltzman's constant. B is the bandwidth, and To is the standard temperature
taken to be 2900K.
The noise factor can be interpreted as the ratio of the actual available output noise
power to the noise power which would be available if the receiver merely amplified the
input noise. This may be expressed as
NF = kToBG+AN = 1 + ANkToBG kToBG (1.1-3)
where AN is the additional noise introduced by the receiver. From equation (1.1-3), AN
can be expressed as
AN = (NF - l) kToBG . (1.1-4)
When two receivers are in cascade, the output noise is due to the sum of the noise
from receiver 1 plus the noise introduced by receiver 2. Let the noise factor for the cascade
and receivers 1 and 2 be denoted by NF, NF1, and NF2, respectively. Then, the output
noise can be expressed as
11
Nout = kToBNFG1G2
= kToBNF1G1G2 + AN2
= kToBNF 1G1G2 + (NF2-1)kToBG 2
Dividing by kToBG 1G2 gives
(NF2 -l)Gl~j (1.1-5)
The effective noise temperature, Te, is defined as that temperature at the input of the
receiver which would account for the noise AN at the output. It follows that
AN = kTeBG = (NF-I)kToBG ,
and
Te = (NF- I)T 0 (1.1-6)
For a cascade of two receivers, let the effective noise temperature of the cascade and
receivers 1 and 2 be denoted by Te, Tel, and Te2, respectively. Similar to the noise factor,
the noise temperature of the cascade is given by
Te = T., + TeG 1 (1.1-7)
If the antenna can be considered to represent a source at the reference temperature,
the equivalent input noise power would be
N = kToBNr.
12
In practice, the antenna will have an effective temperature, Ta which differs from To. The
equivalent input noise at the receiver front end can be expressed in terms of an effective
input temperature Tei which accounts for both the receiver and antenna aoise. Note that
N = k[T(NF-1)+Ta]B = kTeiB.
The effective input temperature is
Tei= T0(NFo-1) + Ta.
The operating noise factor is defined to be
NFO TeiO-To. (1.1-8)
It follows that the equivalent input noise is
N = kT0 B NFo.
With the introduction of NFo, and use of Eqn. (1.1-2), the per pulse signal-to-noise
ratio (S/N)p at the input to the detector assuming no multi-pulse integration becomes
N (4n)3 R4 kTOB NFoL (1.1-9)
where the parameters are:
13
Pt peak transmitted power which is actually the average power during the
pulse duration
G one way antenna power gain (Gt=Gr assumed)
R range to the target
oY target radar cross section
X :wave length at the radar operating frequency
k Boltzman's constant (1.38xl0 -23 W/IHzK)
To standard temperature (2900K)
B receiver bandwidth
NF0 operating noise factor as defined in Eqn.(1.1-8)
L total system loss factor not reflected in the operating noise factor.
The integrated signal-to-noise ratio, (S/N), is the per pulse signal-to-noise ratio,
(S/N)p, multiplied by NI the number of pulses coherently integrated. To account for
integration losses (i.e., the nonideality of the integrator) the loss factor, L, is increased
accordingly. Then,
N' (4X)3 R 4 kToB NF0 L (1.1-10)
If I/B is replaced with the pulse width, 'T, the integrated SNR is equal to the energy
over noise power density, a well known matched filter equation.
When Q, the surveillance volume in solid angle, and TF, the frame time to search
out the surveillance volume are specified, the surveillance radar range equation, (1. I-10)
can be put into a different form. First, the number of antenna beam spots, ns, to cover the
entire surveillance volume is expressed as
]I
where (o is the solid angle subtended by the antenna beam. Note that
A
where A is the effective antenna aperture. In terms of A and X, ns becomes
ns = &
Let Tr and Fr denote pulse repetition interval and pulse repetition frequency,
respectively. PA, the average power, and NI, the number of pulses available for
integration in a beam dwell, are expressed as
PA= _-T = P~
Tr B (1.I-11)
and
N = F[TE FrF 2
n. - A (1.1-12)
The peak power, Pt, can be expressed in terms of the average power, PA, from
equation (1.1-11) as
P = PABFr (1.1-13)
• ,...mnmnmmmnne~mmllII III •nmunn n 1I5
The previously defined expression for the integrated SNR, Eqn. (1.1-10) can be
rewritten by substituting Eqn. (1.1-13) for Pt, and Eqn. (1.1-12) for Ni,
(.)Pt Gz 2 aNI
PAB iA2 2o F F 2-FT X2 1R(4g)3R 4 k To B N FO L GAF/
(PAA)aTF
4iR 4 k To B NF0 L (1.1-14)
PA A in the above equation is called the power-aperture product. Equation (1.1-14)
is an important surveillance radar equation which shows that the integrated SNR is
proportional to the power-aperture product independent of the operating frequency. It
shows that when power-aperture product is fixed, radar performance is fairly well set
unless its frame time and/or the surveillance volume is modified. The effect of varying
these parameters is described in Chapter 5.
In addition to the power-aperture product constraint, performance improvement
must be achieved without losing capability to resolve the range ambiguity inherent in a
High PRF radar, and without exceeding the allowed system false alarm rate.
In a pulse doppler radar, either a range or a velocity ambiguity or both result as the
consequence of a pulse repetition frequency (PRF) selection. The unambiguous range, Ru,
is given by
16
_c(T)
kRu =-(T2
where c is the speed of light and Tr is the pulse repetition interval (PRI). For an
unambiguous range of 400 nautical miles, the PRI, Tr, must be chosen such that
Tr > 2 x 400 x 1852/3x108 = 4.9387 milliseconds.
Hence, the PRF must be 202.4 Hz or lower.
The unambiguous radial velocity, vmax, is given by the doppler frequency
corresponding to one-half the PRF. Note that
PRF/2 = fdmax = 2vmaxA.
Therefore, vmax is determined by the PRF as well as the carrier frequency of the radar
chosen. For unambiguous measurements of both incoming and outgoing target velocities of
up to 1800 knots, the PRF at an operating frequency of 3 GHz should be
PRF > 2ffdmax = 4 x (1800 x 1852/3600)/(3x10 8/3x109) = 37 KHz.
Because of design implementation considerations some radars use PRF that is equal to just
the maximum expected doppler frequency of the target. This results in an ambiguity as to
opening or closing of the target velocity.
The superiority of a High PRF radar for detecting high speed airborne targets over a
heavy ground traffic environment has been well demonstrated as was shown in the perfor-
mance comparison between E-3 AWACS and E-2C Hawkeye radars over the European
17
continent. The E2-C radar operating at p-band (UHF frequency) has a broad antenna
beam. With its low PRF waveform the largest mainbeam clutter doppler spread due to
platform motion when the antenna beam is pointed normal to the platform velocity vector
occupies almost 30% of the PRF intervaL This PRF approximately corresponds to a 200
knot target speed. Thus, the radar cannot see targets whose speeds fall in this blind velocity
zone which can be alleviated somewhat by PRF staggering. Of course, high PRF radars
suffer from eclipsed range. The real difficulty with the overland performance of the E-2C
radar, even with its scan-to-scan processing to suppress detections from ground vehicles
and discrete land clutters, is its inability to distinguish slow moving ground vehicles from
the high priority airborne threats in the portion of the PRF interval where the radar is not
blinded. The high PRF radar is much more complex to build, however.
The range ambiguity in a high PRF radar is resolved by the so called Chinese
remainder theorem [3, p19-16]. This approach permits a unique direct computation of the
true range cell, Rc, from the multiple ambiguous range cell numbers, Al, A2, -, An. In
particular, for the three PRF system Rc is given by
Rc = (CIAI + C2A2 + C3A3) modulo(mlm2m3)
where the mi's are the number of range cells in each PRI and are required to be relatively
prime. That is, Rc is the remainder of the term within parentheses, when divided by
mlm2m3 as many times as possible. The constants C1, C2, and C3 are related to ml, m2,
and m3 by the following equations:
ClI = b 1m2m3 =(1) modulo (m 1)
C2 = b2mlm3 = (1) modulo (m2)
18
C3 = b3mlm2 = (1) modulo (m3)
where bl is the smallest positive integer which, when multiplied by m2m3 and divided by
ml, gives unity as a remainder (and similarly for the other b's). Once ml, m2 and m3 are
chosen, the range, Rc, can be computed by using the C values and the ambiguous cell
numbers (A1,A2,A3) in which the target is detected. For example, the Chinese remainder
theorem says that, given a triplet whose elements' maxima are 3, 5, and 7, any number
between 1 and 105 can be uniquely specified by the triplet, ( 1,1,1) corresponding to 1 and
(3,5,7) corresponding to 105.
To summarize, resolving range ambiguities in a High PRF system requires more
than one coherent processing interval (CPI) in an antenna beam dwell with the attendant
time overhead associated with each CPL This time overhead is 50% of the available time in
the example radar. This is why it is impractical, unless the scan rate is reduced to increase
the dwell time, to have more than 3 CPrs in a beam dwell even though, theoretically, more
CPI's can lead to improved performance of the binary post detection integrator.
Lastly, detection performance improvement is unacceptable if the false alarm rate is
not kept below the minimum desired or tolerable level. Therefore, needless to say, the
specified false alarm rate must not be exceeded under any condition. According to the
preferred Neyman-Pearson strategy, the improvement should be in terms of maximizing the
probability of detection for a fixed value of a false alarm probability that maintains the
system false alarm rate at or below the allowed value.
19
1.1.3 Performance Improvement Criteria
Performance of the radar using the proposed modifications is compared to that of its
present (baseline) configuration. A cumulative detection probability of 0.9 in a one minute
time span for Swerling case 1 targets in the entire surveillance volume is adopted as the
Principal performance criterion under the constraint that the minimum false alarm time is 5
seconds. All suggested modifications are made within the system constraints specified in
Section 1.1.2, and the Gaussian assumption for the noise and signal plus noise is assumed
valid such that the receiver structure shown in Figure 1.1-2 is the optimum receiver which
maximizes detection probability.
The Gaussian assumption is certainly true for the Swerling and Marcum target
models in white Gaussian noise considered in this report where target doppler
frequencies fall in the clutter free doppler zone after target signals are separated from the
mainbeam clutter by their doppler frequencies and the mainbeam clutter is removed. This
is also true in the case where target doppler frequencies coincide with doppler frequencies
of sidelobe clutter. The reason for this is that the sidelobe clutter level in this radar is equal
to or less than the thermal noise due to the extremely low sidelobes of the antenna. Also
the sidelobe clutter is the sum of clutter returns from all ground patches whose doppler
frequency and range rings fall in the ambiguous radar range-doppler cell in question such
that the central limit theorem applies.
In the edge of the mainbeam clutter and in altitude.line clutter, the clutter signals can
be greater than thermal noise. Even here, experimental data shows that the marginal
probability density of clutter signals received through a coarse resolution radar is Gaussian
within the time and spatial limit bounded by a single range-azimuth resolution cell in a
beam dwell wherein a detection decision takes place. That is, even though the spatial
inhomogeneity of the terrain gives rise to non-Rayleigh density functions for the amplitude
20
distribution of the clutter taken over a large surveillance area, the conditional density
function given the local mean of the clutter voltage returning from many elemental clutter
cells contained in a radar range-azimuth resolution cell is Gaussian by the central limit
theorem. This random process, however, is characterized by nonzero mean and unequal
variances between the quadrature components, and with a high degree of pulse to pulse
correlation. This is believed to be because the sample is taken from a ground patch within a
clutter spatial correlation distance and within a temporal correlation time. This is the non-
stationary characterization of the clutter. More thoughts on this and some performance
comparisons of several filter designs intended to maximize output signal-to-clutter ratio are
presented in chapter 6.
Returning to the performance criterion, cumulative detection probability, Pc, is an
important performance measure of a surveillance radar. Its definition and its relationship to
the single scan probability of detection, otherwise known as the blip-scan ratio, will now
be discussed. Consider the surveillance volume centered about the radar.platform shown in
Figure 1.1-3. The outer boundary of the horizontal coverage is defined by the range
beyond which a single scan probability of detection, Pd, is less than some minimum value.
Consider a radially approaching target with velocity VR. Once the target has penetrated the
boundary of the surveillance volume, let AR and At represent the maximum range
penetration and the maximum time elapsed, respectively, before the target is detected by
the radar at least once. Obviously, AR and At are related by
AR = VRAL
In general, several scans (say L) occur while attempting to detect the target in the interval
AR. For analytic convenience, assume that AR is small compared to the range at which AR
is measured such that the single scan detection probability (Pd) within the time At can be
21
radar
for radially approaching targetwith velocity Va
AR Vut
Horizontal Coverage
round earth
Vertical Coverage
Figure 1.1-3 Surveillance Volume for an Airborne Radar
224
assumed to be constant. With this assumption and the assumption of statistical
independence from scan to scan, Pc in time At is given by
Pc = 1 - (1 - Pd)L . (1.1-15)
In equation (1.1-15), Pd is the single scan probability of detecting the target. Thus,
(1 - Pd) represents the probability of a miss in any one scan and (1 - pd)L is the probability
of failing to detect the target in all L scans. Hence, Eqn. (1.1-15) gives the probability of
making at least one detection in L scans.
On the surface, examination of Eqn. (1.1-15) indicates that Pc can be increased by
using a larger value of L. However, for a specified At,
L = At 'F
where TF is the frame time allowed for searching out the entire surveillance volume. When
At is held fixed, the only way to increase L is to decrease TF. However, a reduction in TF
results in a shorter dwell time which, in turn, decreases NI, the number of pulses to be
integrated. Consequently, Pc becomes smaller as L becomes larger. It follows that there
exists an optimum value for L in Eqn. (1.1-15).
A second consideration arises for fluctuating targets. It is shown in Section 3.3
that, as the signal-to-noise ratio increases, the detection probability for a fluctuating target
relative to that of a nonfluctuating target degrades as shown in Figures 3.3-1 and 3.3-9.
For a Swerling case 1 target, a crossover occurs when Pd = 0.32 in a single slant system or
if an average beam shape loss is assumed for all slants in a multi-slant system. Otherwise,
the crossover occurs at a point where Pd is slightly less than 0.32. For Pd less than this
23
value, the fluctuating target has a higher detection probability than does a nonfluctuating
target. The situation is reversed for Pd greater than this value. Barton [4] states that the
optimum number of scans, L, for a Swerling case 1 target is around 6 for Pc of 0.9. It is
around 2 for Pc of 0.5 and around 10 for Pc of 0.99. For the radar under consideration in
its baseline configuration, the specified minimum Pd is 0.32 and the maximum elapsed
time At is one minute. This gives 6 scans in At at a scan rate of 36 degrees per second and
Pc in At becomes
Pc = 1 - (1 - 0.32)6 = 0.9 (1.1-16)
This satisfies the optimum number for L of Barton.
If the range closure during the elapsed time At is not negligible, then, a different Pd
for each scan must be used to compute Pc. Let AR = LA where A is the range closure
during a single scan. Assuming a constant radial velocity, the target in a particular azimuth
drection appears in the same relative position within each increment A, as shown in Figure
1.1-4. Let p(Rm+r+iA) denote the single scan Pd at range Rm+r+i& However, depending
on the azimuth direction, r can vary anywhere from Rm to Rm+A. Assuming r is a random
variable uniformly distributed over the interval A, the mean cumulative detection
probability, P(Rm, A) is given by
P(Rm,A)- 1f - [l-p(Rm+r+iA) dr.
(1.1-17)
The integral in Eqn. (1.1-17) performs an averaging which takes into account the fact that
the target could be anywhere in the interval [iA, (i+l)A] with equal likelihood when
24
intercepted by the radar beam. Analyses based on this approach which allow determination
of the optimum value of frame time is given in Sections 5.1 and 5.2. Unless otherwise
specified, the definition for Pc given by Eqn. (1.1-15) will be used in this investigation.
ion~-~ zoneF 1lativeetection Tet on
Ct l ,_ .n ta r ge t
aqProaC '
radar pl t o r AVR
Rm+,& am+ (L-1)
Figure 1. 1 -4. Detection Scenario of an Approaching Target with a Constant Radial Velocity
The relationship between Pd and Pc has been established when Pd is independent
from scan to scan. As will be shown in Section 4.4, this assumption is not valid when
scan-to-scan processing (SSP) is used. Therefore, for SSP, Pc cannot be computed by
simply using Eqn. (1.1-15). The computation of PC for SSP is developed in Chapter 4
which is entirely devoted to SSP.
Another term often used in radar detection is the blip-scan ratio (BSR). This is the
ratio of the number of scans, J, within each of which a given target is detected to the total
number of elapsed scans, L. For L large, the blip-scan ratio approximates the single scan
detection probability in accordance with the relative frequency interpretation of probability
[5]. Note that the average number of hits in L scans can be approximated by the product
25
(BSR]L. This is also equal to the number of track updates in a time interval equal to L
scans.
The track update rate at the minimum critical range is another performance criterion.
This is the range at which the single scan detection probability is high such that the track
update rate approaches the antenna scan rate. By definition, the track update rate is the ratio
of Pd to TF. The track update interval is the reciprocal of the track update rate. For
successful target tracking, a minimum track update rate must be maintained at the minimum
critical range. This must be kept in mind when adjusting scan rate so as to improve
cumulative detection performance. Scan rate optimization is discussed in Chapter 5 where
it is shown that the cumulative detection probability can be increased by slowing the scan
rate. However, the track update requirement places a lower limit on scan rate.
1.2 Dissertation Outline
The description of the radar for which performance improvement is sought was
presented in Section 1.1.1. Chapter 2 contains a discussion of various considerations
needed for comparison of the baseline and modified radars. These include such issues as
the number of coherent processing intervals in a beam dwell, the number of available
pulses in a coherent processing interval, beam shape loss, false alarm probability
allocation, and target models. It is shown how two different approaches to false alarm
calculations, as proposed in the radar literature, can be used to relate cell false alarms to
system false alarms.
Performance analysis of the baseline radar is presented in Chapter 3. This forms
the basis for comparing performance improvements of the modified configurations. It is
desired to extend the range at which the specified cumulative detection probability is
achieved without changing the radar's power-aperture product and operating frequency.
26
Scan-to-scan processing has been pursued by others as a promising technique for
obtaining performance improvement. However, a detailed analysis to support this claim is
not available in the literature. In Chapter 4, a careful analysis is performed and numerical
results are generated for two versions of the scan-to-scan processing concept It is shown
that only marginal improvement can be attributed to scan-to-scan processing. One
significant conclusion is that it is difficult to overcome the power-aperture product
limitation with increased signal processing.
In an effort to overcome the limitations encountered, an alternate approach is
explored in Chapter 5. This involves scan rate reduction and non-coherent integration in
place of the M of N post detection integration. Trade-offs involved with reducing the search
sector are also examined because the power-aperture product constraint leaves very few
options. A detailed analysis reveals potential improvements over the range from 3 to 10 dB
in the equivalent signal-to-noise ratio gain. The near 10 dB improvement occurs with
reduction of the search sector by a factor of two, slowing down the scan rate by the same
factor coupled with a non-coherent integration in lieu of the binary post detection
integration, and modifying the performance requirement that Pc=0.9 in a 1 minute
surveillance interval to the requirement that there be on the average one track update in a 10
second interval. The improvement, strictly according to the original performance criterion,
can be as high as 5 dB. This requires noncoherent integration with slant-to-slant frequency
agility. All these figures are for Swerling case 1 targets.
A summary of results developed in this dissertation is presented in Chapter 6. The
work is focused on improving detection of targets embedded in white Gaussian noise. In
addition, some thoughts on clutter statistics and their implication on receiver structure are
given. Quantitative comparisons of relative performances of different classes of clutter
27
cancellers using actual high PRF radar data as test inputs are also included. Finally,
suggestions for additional work are included.
28
CHAPTER 2
SOME CONSIDERATIONS IN PERFORMANCE COMPARISON OF
MODIFIED AND BASELINE RADAR CONFIGURATIONS
In this chapter various concepts needed for comparison of
the baseline radar configuration with proposed modifications
are discussed. After defining relevant radar terminology,
assumptions used in the analyses are presented. This is fol-
lowed by discussions of those parameters which do not remain
constant as the configuration is changed from the baseline.
2.1 Definitions of Terms
To facilitate understanding of the discussions and
analyses presented in this investigation, frequently used
terminology which may not be familiar to the nonspecialist
is defined below:
Baseline Configuration:
The radar in its present form is referred to as the
baseline configuration to distinguish it from modified con-
figurations proposed for performance improvement.
Surveillance Volume (Q):
Surveillance volume is the volume of space to be searched
29
for targets of. interest. The surveillance volume of this ra-
dar was shown in Figure 1.1-3. The azimuth coverage is 360
degrees. For range less than Ri, the vertical coverage is
bounded by the antenna elevation beamwidth while beyond Ri,
the coverage is bounded by the surface of the earth and the
maximum altitude of interest. Both coverages are bounded in
range by the radar horizon and/or the maximum detection ca-
pability of the radar.
Search Frame Time (TF):
For the radar under consideration the vertical coverage
is illuminated by the single antenna beamwidth. Therefore,
the antenna scan is limited to the azimuthal dimension. The
search frame time is the time for the antenna fan beam to
sweep through the entire 360 degree azimuth sector. The ra-
dar under consideration is equipped with a mechanically ro-
tating antenna. The search frame time is more commonly re-
ferred to as a frame time or a scan time. For the radar in
the baseline configuration, the scan time is 10 seconds.
Therefore, the scan rate is 36 degrees per second or 6
revolution per minute (rpm).
Beam Spot:
For a given antenna beam, the number of beam spots re-
quired to cover the entire surveillance volume determines
30
the number of beam dwells in a scan and, therefore, the
dwell time that results for a specified frame time. In an
azimuth scan only radar, there are 360 0/aB beam spots in a
3600 sector where eB is the one-way half-power azimuth
beamwidth.
Dwell Time:
For the radar under consideration the antenna is me-
chanically rotated in azimuth at a constant rate. The dwell
time is the time during a single scan that a point target in
a fixed azimuth direction is within the one-way half-power
beamwidth.
Pulse Repetition Frequency (PRF):
The pulse repetition frequency is the frequency of the
periodic transmitted pulse train. When several different
PRF's are used in order to resolve range ambiguities, the
number of pulses transmitted in a given time, assuming con-
tinuous transmission at an average PRF, is used to measure
time intervals of interest, e.g., dwell time, for conve-
nience.
Coherent Processing Interval (CPI):
CPI is the time interval spanning the number of pulses
used for coherent integration (i.e., number of FFT samples).
31
The duration of the CPI is measured in terms of the average
pulse repetition frequency (PRF). CPI's in a dwell time are
not contiguous. They are interleaved with intervals for time
overhead.
Time Overhead (TH):
Time overhead is the portion of the time in a modulation
period that is not a part of the coherent processing inter-
val. This includes the round trip transit time of the trans-
mitted pulse with respect to the maximum range clutter patch
and/or a target, certain house keeping overhead such as tim-
ing and control necessary to setup the CPI, and clutter can-
celler transient response settling time during which signal
integration is prevented. In the baseline configuration,
this time overhead exceeds 50% of the modulation period.
Modulation Period (Tm):
A frame time or a scan time consists of a contiguous
train of modulation periods. A modulation period consists
of the coherent processing interval (CPI) plus the total
amount of time overhead associated with the CPI. The modula-
tion period is depicted in Figure 2.1-1. The actual received
pulses at the analog-to-digital converter output are shown
in Figure 2.1-2 for range gates 33 through 36 where the in-
teractions between first, second and multiple time around
32
modulation period
leading edge trailing edgehouse keeping house keeping
transmit fffl~ffi I
echo from echo from
echo fro min echo from minRmax Rax
receive _ IK4multiple time around echo
settling time
process _'_ __ _ __
clutter canceller CPItransient period
Figure 2.1-1 Modulation Period
33
9L t I>
o Ito
C cc a
-.'0 a 0-
higdj opqgd&
34.
echoes are clearly illustrated at the beginning of the pulse
train. These pulses are discarded from further processing.
This transient period lasts until returns from the clutter
patch at the maximum range begins to reach the receiver.
Figures 2.1-3 (a) and (b), respectively, show the output of*
the infinite impulse response (IIR) filter used for the
clutter canceller and the steady state portion of that
waveform processed through a 128 point FFT. Figures 2.1-3
(c) and (d), respectively, are the output of the finite im-
pulse response (FIR) filter used as the clutter canceller
(2-stage delay line canceller) and that waveform processed
through a 256 point FFT. The absence of the transient re-
sponse at the output of this FIR clutter canceller is
evident and is what enables use of the longer FFT.
Slant:
A slant is synonymous with modulation period. A 2-slant
or 3-slant configuration refers to the configuration wherein
a dwell time is divided into the specified number of slants.
Integrated Signal-to-Noise Ratio (SNR or S/N):
As given in Eqn. (1.1-10) the integrated signal-to-noise
ratio involves the average signal power over N pulses.
Some authors use peak signal power over the NI pulses. Con-
sistent with DiFranco and Rubin [2] the symbol,9?, will be
employed to denote the integrated signal-to-noise ratio when
35
U.
..
. -I.,
_
41~
.0.3
u o4
0 0
_ _ j 0. >-
no 'C 0
--------- cow0.0
U aJ4 r1.U C0
a%_________________ t I I I I
06 3 6
peak signal power is used. Note that W is twice SNR. Un-
less otherwise specified, the SNR in Tables and Figures of
detection probability (Pd) will be the SNR associated with a
single CPI that would appear at the detector had all the
pulses in that CPI been received through the peak of the an-
tenna beam. Hence, the notation (S0/N) is introduced to
signify this. When this is in response to an average target
radar cross section (RCS), an overbar is placed above the
symbol, i.e., (S /N). To take into account the beam shape,
the Pd is computed by subtracting from the listed (S0/N) the
corresponding beam shape loss in dB for each of the CPI
within a beam dwell in a given processing configuration.
This allows for comparison, on a common ground, of the dif-
ferent processing configurations which have different beam
shape losses.
Reference Range (R0):
Reference range is the range at which the specified de-
tection performance is reached for a reference Swerling case
1 target in the baseline configuration. All other ranges are
normalized to this range.
Reference Signal-to-Noise Ratio (9/N)O:
Reference signal-to-noise ratio (reference SNR) is the
integrated signal-to-noise ratio received in a CPI from a
37
reference target having a specified mean radar cross section
(RCS) when it is located at the reference range. All refer-
ence targets have the same specified mean RCS. When the ref-
erence target is a Swerling case 1 target, the reference SNR
results in a single scan detection probability of 0.32 in
the baseline configuration. Since the reference target is
defined only in terms of its mean RCS regardless of the tar-
get model assumed, the reference SNR has the same value for
all target models in a given configuration but gives rise to
different values of probability of detection. For configura-
tions other than the baseline, the reference SNR is the SNR
obtained in a CPI in each configuration from the reference
target located at the reference range. Therefore, the value
of the reference SNR is different for a different con-
figuration and thus reflects the change in the configura-
tion.
Single Scan Detection Probability (Pd):
By definition, single scan detection probability which is
measured after post detection integration, does not utilize
information from previous scans. Therefore, Pd is assumed
to be independent from scan to scan.
Hybrid Single Scan Detection Probability (Pd(hyb)):
In contrast with single scan detection probability, the
38
hybrid single scan detection probability which arises in SSP
uses the past scan detection history. Therefore, it is not
independent from scan to scan.
Blip Scan Ratio (BSR):
Blip scan ratio is the number of scans, J, where a hit is
scored from a given target divided by the total number of
scans, K, elapsed. This approaches the single scan detection
probability when K is large assuming that Pd remains con-
stant.
Cumulative Detection Probability (P,):
Cumulative detection probability and its relationship to
single scan detection probability (P d are defined in
section 1.1.3 under performance improvement criteria. Unless
otherwise specified, cumulative detection probability over a
fixed time interval is defined assuming that single scan de-
tection probability is constant over that interval. This, in
turn, assumes that the target range closure during the in-
terval is negligible. Thus, in the absence of a scan-to-scan
processing (SSP), Pc is given by the relationship,
Pc 1-(1-Pd )L , where L is the number of antenna scans in that
time interval and Pd is assumed to be independent from scan
to scan. The computation of Pc under SSP is complex. This is
derived in chapter 4 where SSP is discussed. Unless
39
otherwise specified, the time interval is taken as one
minute or L is 6 scans at 6 a rpm antenna rotation rate.
2.2 Basic Assumptions
Basic assumptions under which the results of this inves-
tigation are based are presented in this Section.
1. The probability density of signal and signal plus noise
voltages after removal of the mainbeam clutter is
Gaussian. The noise voltages which may consist of the
thermal noise and residue mainbeam clutter plus sidelobe
clutter are white Gaussian with zero mean and equal vari-
ance between their quadrature components.
2. The antenna and the receiver together, including
analog-to- digital converter and FFT before the detector,
is linear.
3. The target model of interest belongs to one of the five
Swerling and Marcum cases. In particular, the priority
target is Swerling case 1.
When assumption 1 does not hold, particularly with respect
40
to the total noise remaining after the mainbeam clutter can-
cellation, the receiver structure shown in Figure 1.1-1 is
not optimum for maximizing the detection probability. Avail-
able test data shows, however, that the stated assumption
is reasonable.
41
2.3 Number of CPI's in a Beam Dwell and Number of Available
Pulses in a CPI
As introduced in section 2.1 under definition of terms,
the frame time of a radar whose antenna is scanning the sur-
veillance sector at a constant rate is made up of contiguous
modulation periods. The length of a modulation period is
chosen such that there can be at least 3 modulation periods
in a beam dwell in order to allow use of different PRF's to
resolve the range and/or velocity ambiguities of targets.
Associated with each modulation period, there is a fixed
time overhead. For the baseline configuration, this is over
50% of the total modulation period. In general, for a
specified beam dwell, this limits the maximum number of
modulation periods in the beam dwell to 3. (For other ra-
dars, in which a smaller time overhead is possible primarily
because of shorter round trip transit time for surveillance
over short ranges, more modulation periods can be imple-
mented for greater effectiveness in binary post detection
integration.)
Given a system which is power limited (i.e., a transmit-
ter which is already operating at the limit of its peak
power, average power, and duty factor), the most effective
operation uses the widest pulse width at its peak power con-
42
sistent with resolution requirements, and the maximum PRF
for which the average power and duty factor are not ex-
ceeded. This, in essence, sets the maximum PRF as well.
Thus, the available energy in a beam dwell is fixed for a
given scan rate and pulse repetition frequency. However,
usable energy is determined by the number of modulation pe-
riods chosen and the fixed time overhead associated with
each modulation period. The relative figures based on these
numbers are established here for the baseline and modified
configurations for convenient reference.
In practice, different pulse repetition intervals (PRI's)
are used in the modulation periods which reside in a beam
dwell. In this investigation, relative time intervals are
measured using the average PRI. For example, consider a
3-slant configuration in which the PRI's are 0.9, 1.0 and
1.1 milliseconds. (These are fictitious numbers.) Conse-
quently, a time interval of 1 second will be said to have a
duration of 1000 PRI's using the average PRI. Analogously,
since there is one pulse per PRI, time can be measured in
terms of the number of pulses. A dwell time is defined as
dwell time = (half power beamwidth)/(scan rate).
Let PRI and PRF denote the average pulse repetition interval
and average pulse repetition frequency, respectively. Of
43
course, PRF is the reciprocal of PRI. A dwell time is given
in terms of a number of PRI's by
dwell time (in PRI's) = (dwell time)/PRI
- (dwell time)xPRF.
For a particular radar design, the number of modulation pe-
riods processed as a group will occupy a specified fraction
of the beamwidth. (This fraction typically varies between
0.8 and a number slightly higher than unity.) Thus, an av-
erage modulation period measured in PRI's is given by
modulation period (in PRI's) = (dwell time)x(fractional
beam width used)xPRF / (number of modulation periods).
In practice, [(dwell time)x(fractional beamwidth used)] is
also referred to as dwell time for convenience.
A coherent processing interval (CPI) available in a
modulation period is the time remaining after a fixed time
overhead is subtracted from a given modulation period. Two
different time overheads are considered in the performance
analysis: One is the normal time overhead (also referred to
as the maximum time overhead), and the other is a reduced
time overhead. The latter is based on the assumption that
under some operating conditions, where there is no clutter
beyond the radar horizon, the waiting period for the mul-
tiple time around echo from the farthest clutter patch can
44
be reduced. Another factor influencing time overhead is the
transient settling time of the clutter canceller. It appears
possible to reduce time overhead by incorporating a cancel-
ler having a shorter settling time. The expected savings
arising from the two factors are incorporated into the re-
duced time overhead. The relative figure proportional to the
number of pulses available for integration, along with re-
lated parameters, are listed in Table 2.3-1 for a radar
employing a mechanically rotating antenna with a 6 rpm scan
rate.
It is assumed that N B number PRI's occur within the an-
tenna half-power beamwidth. Because 103.6% of the beamwidth
is utilized, the total number of PRI's in a beam dwell is
1.06NB. Consequently, there are 1.03 6NB/3 = Nm3 number of
pulses in a modulation period for a 3-slant configuration
and 1.036NB/2 = Nm2 number of pulses for a 2-slant con-
figuration. Because of the large time overhead, the number
of useful pulses in a CPI is significantly smaller than the
total number of pulses in a modulation period. The reference
number of pulses per CPI is equal to (reference NI). The
relative integration gain per CPI for the various configura-
tions is defined to be
relative integration gain (dB) # of pulses in a CPICPI = 10log10 (reference NI)
45
Table 2.3-1
Number of Pulses Available for Integration witha 6 rpm Antenna Scan Rate
Configuration 3-slant 2-slant
Time overhead normal reduced normal reduced
No. of pulses in3-dB beam width N B N B N B N B
% beam widthutilized for PDI* 103.6 103.6 103.6 103.6
No. of mod. periodsper beam dwell 3 3 2 2
No. of pulses in amod. period (MP)** Nm3 Nm3 Nm2 Nm2
time overhead .521N .348Nm3 .347Nm2 .232Nm2
CPI .479N .652Nm3 .653Nm2 .768Nm2
Relativeintegrationgain/CPI (dB) Ref 1.33 3.11 3.81
Relativegain in totalenergy/dwell (dB) Ref 1.33 1.35 2.05
Baseline Modified Configurations
* PDI = post detection integration
** Nmi = (NB)(% beamwidth utilized)/(i)
46
The reference number of pulses per beam dwell is 3x(refer-
ence NI). Hence, the relative gain in total energy per
beam dwell is given by
relative gain in total energy (M of pulses in a CPI)(# slants)beam dwell (dB) = 10 log10 3x(reference NJ)
The relative integration gain/CPI is a misleading figure
of merit because the larger gain for a 2-slant configuration
is offset by the fact that there is one less CPI than for
the 3-slant configuration. Both of these factors determine
performance of the M of N binary post detection integrator.
As a result, the relative gain in total energy/beam dwell is
a figure of merit which more closely reflects system perfor-
mance.
2.4 Beam Shape Loss
Broadly speaking, beam shape loss takes into account the
variable antenna gain experienced by the transmitted and re-
ceived pulses as the antenna scans by a point target during
a CPI. Blake [7] has shown that a minimum two way, single
dimensional beam shape loss of 1.6 dB should be used to ac-
count for the actual beam shape as compared to the constant
gain over the half power beam width typically used. His
47
analysis assumes that a large number of pulses are centered
around the peak of the beam over an optimum fraction of the
beamwidth. For pulses positioned outside the 84 % beamwidth,
he shows that integration results in loss of signal-to-noise
ratio because more noise is added than signal beyond this
optimum width. His analysis applies to a detection of a weak
signal with a square-law detector followed by a post detec-
tion integrator.
Since finding an appropriate average beam shape loss de-
pends on the percent of beamwidth utilized and the par-
ticular detection processing used, a more accurate method is
to compute the actual loss per CPI. Let the two-way antenna
power pattern be approximated by the Gaussian expression,
G( ) - exp(-5.55 0/) .
Assuming a train of pulses at a constant PRF, let N B denote
the number of pulses that would be received in a beamwidth.
If the pulses are centered about the peak of the beam and
0k denotes the angular position of the kth pulse, it follows
that
9k kO -N k (2.4-1)
O- NB
48
Let NI be the number of pulses actually integrated. For the
idealized situation in which the beam pattern has a constant
gain of unity, the total received power would be N times
the power in a single pulse. Assuming the N, pulses to be
centered on the beam where NI is an odd integer, the normal-
ized total power received is
(N1 -1)/2
1 + 2 E exp(-5.55 k/NB)k=1
The beam shape loss, LBS , is defined to be the ratio of
these two powers, expressed in dB. Hence, as found in
Skolnik [9], the beam shape loss is given by
N1LBS w 10 log10 (N-1)/2 (2.4-2)
1 + 2 E exp( -5.55 kfNB2)k=1
For example, if there are 11 pulses to be integrated, all
lying uniformly between the half-power beamwidth, the beam
shape loss is 1.96 dB.
In the radar under consideration the set of modulation
periods (e.g., 3 in 3-slant configuration) nearest the beam
center is selected as a group, at a one slant increment on a
sliding window basis, for post detection integration. The
set nearest the beam center will result in the highest
signal-to-noise ratio. Therefore, this particular set is the
one for which the beam shape loss needs to be determined.
49
Since the antenna is continually scanning, the set can be
positioned anywhere with respect to the beam center with
equal likelihood from the most favorable position (centered)
to the least favorable position which is one-half modulation
period offset left and right from the centered position. For
an offset larger than this, the next set will result in a
higher signal-to-noise ratio. Therefore, the correct beam
shape loss for each CPI in a set is the beam shape loss for
the CPI computed per Eqn. (2.4-2) adjusted for offset from
the beam center. In addition, to account for the random po-
sition of the CPI, the beam shape loss should be averaged
over all possible positions of the CPI extending from
one-half modulation period offset to the left and right. The
centered and extreme offset positions for 2-slant and
3-slant configurations are shown in Figure 2.4-1.
Analogus to Eqn. (2.4-1), it is convenient to measure an
angular offset in terms of the number of pulses that would
span the offset or length. In this sense, the following no-
tation is introduced (see Figure 2.4-1):
N : offset of a CPI measured from the beam centerlinec to the center of the CPI when the set of modula-
tion periods is centered on the beam centerline
NL : offset of a CPI measured from the beam centerlineto the leading edge of the CPI
NI : length of the CPI
50
2-Slant System:
C. Beam
Mod Period,
Centered
V - Mod Period
I I/ I Left Offset
I -- Mod Period
I FI I Right OffsetI I I
3-Slant System:• CL Beam
Pwr Beam Width (N) - J/ __~_+j( NMod)f__ -NL - "NCPI (NJ )
,/CNI ., 'Centered
Mod Period
V/I/Il 1//f//ALeft Offset
MUM Right Offset
Figure 2.4-1 Possible Positions of a Set of Modulation
Periods w.r.t Beam Centerline
51
Nmod : length of modulation period.
For a CPI offset by NL, the conditional beam shape loss of the
CPI is
L9~ - 0 081 [NI+LE exp(-5.55k/N)]
To average over the random position of the CPI, let NL be
uniformly distributed over the modulation period. It follows
that the beam shape loss, averaged over all possible posi-
tions of the CPI, is given by
IBS = 10 1lo N1+NL-1 N
D=-j(N..dGN)+NC E exp(-5.55k2/N B (2.4-3)
In the remainder of the investigation the term, beam shape
loss, refers to the loss evaluated using Eqn. (2.4-3).
Using this method, the beam shape loss for each CPI in
various 2-slant and 3-slant configurations are computed. The
results are summarized in Table 2.4-1. Because of symmetry
in the 2-slant configuration, the beam shape loss for both
CPI's are equal. For the 3-slant configuration the loss for
the center CPI is significantly less than those for the two
outer CPI's. The losses with reduced time overhead are
about the same as those for normal time overhead. Almost
52
without exception, the industry computes performance
analysis using the 1.6 dB beam shape loss originally pro-
posed by Blake [7]. When several slants are involved, the
1.6 dB loss is treated as though it applies to each of the
individual CPI's. In Table 2.4-1, average beam shape losses
are tabulated which are obtained by averaging the actual
losses for each CPI. The increase in loss cver the 1.6 dB
'.gure is due to both the larger percentage of beam utiliza-
tion and the averaging over the random positioning of the
CPI's. in the analysis of the baseline configuration, the
effect of beam shape loss on the detection probability is
discussed. It will be shown that use of an average beam
shape loss in place of the losses for individual CPI's
leads to an optimistic prediction of performance. As a fi-
nal point, it is noted that the use of additional slants can
be achieved by processing CPI's positioned outside the
half-power beamwidth. The beam shape loss for those CPI's
increases dramatically. For example, if 5 slants in a
3-slant configuration are processed, the oeam shape loss for
the two outer CPI's in the 5-slant set is 7.28 dB. There-
fore, use of Blake's 1.6 dB figure for those CPI's would re-
sult in large errors.
53
Table 2.4-1 Summary of Beam Shape Loss for the 2-Slantand 3-Slant configurations
2-slants per beamwidth 3-slants per beamwidthbeam shape loss (dB) beam shape loss (dB)
CPI 1 CPI 2 (2-CPI) ave. CPI1 CPI2 CPI 3 (3-CPI)ave.
normaltime 2.55 2.55 2.55 3.43 0.30 3.43 2.12overhead
reducedtime 2.52 2.52 2.52 3.39 0.34 3.39 2.12overhead
54
2.5 False Alarm Probability Allocation and System False
Alarm Verification
A system false alarm occurs when the target report gen-
erator following the post detection integrator erroneously
declares the presence of a target. On the other hand, a
cell false alarm before post detection integration results
when the test statistic for that cell exceeds the threshold
in the absence of a target. The allocation of a cell false
alarm probability such that the probability of system false
alarm remains below an acceptable level during actual radar
operation is a complicated procedure for modern pulse
doppler radars having range and/or velocity ambiguities
which are resolved by post detection integration of multiple
observation samples. A specification of cell false alarm
probability, which is frequently done in practice, does not
lead to a specific system level false alarm performance un-
less the underlying detection process is also specified.
Two competing processing schemes should not be compared us-
ing the same cell false alarm probability because they would
in general lead to different system false alarm performance.
Marcum (see Nathanson [10j) did the first work in false
alarm calculation. Let Pf denote the false alarm probability
55
each time there is an opportunity. On the average, assume
there are n' independent false alarm opportunities in a time
T'fa* Let P denote the probability of no false alarm in n'
false alarm opportunities. Then, P0 is given by
Po 'A (1 - Pf)' (2.5-1)
If a denotes the average number of independent false
alarm opportunities per second, then, n' is given by
nq ,= aTfl.
Eqn. (2.5-1) can then be written as
Po = (1 - Pf)ea fa. (2.5-2)
Given values for Po, Pf, and a, Marcum defined the false alarm time to be
that value of Ti. such that Eqn. (2.5-2) is satisfied.
For large n', Eqn. (2.5-1) can be approximated by
-n' PfP0 = e
It then follows that
56
P - In- ". (2.5-3)Pf n' TO.
Marcum selected a value of 0.5 for P0 With this choice,
the probability of having no false alarm is also equal to
the probability of having one or more false alarms. Conse-
quently, a complementary equation to Eqn. (2.5-1) is
0.5 = I - (I - pf)n'. (2.5-4)
Substituting Po = 0.5 into Eqn. (2.5-3) gives
69 0.69Pf -n W-F. (2.5-5)
An alternate approach was proposed by Barton (4] and
Skolnik (9]. Let na be the number of false alarms in time
t. They defined the average false alarm time, Tfa , as
Tf urn _f, - t (2.5-6)t-.r'o a
As before, let denote the average number of independent
false alarm opportunities per second. Then, the average
number of false alarm opportunities in time t is equal to
t. From the relative frequency definition of probability,
57
the false alarm probability is given by
Pf r lim a - in1nfl (2.5-7)
From Eqn. (2.5-6), it follows that
P (2.5-8)
Equating Eqns. (2.5-5) and (2.5-8), it is seen that
Tf. = 0.69 Tt, . (2.5-9)
provided Po = 0.5 and n' is large.
Now a method for deriving a cell false alarm probability
before post detection integration from a specified system
false alarm rate requirement is developed. This is done
first by using Marcum's definition for the 3 of 3 post de-
tection integration in the baseline configuration of the ra-
dar under examination. A method for general M of N post de-
tection integration is then derived using the Barton and
Skolnik approach but suitably modified to take into account
the effect of binary post detection integration, time utili-
zation, range eclipsing, velocity blanking, and
range/velocity unfolding. Finally, this is specialized for
58
the 3 of 3 processing to show that the two methods give re-
sults having the same order of magnitude.
In 3 of 3 post detection integration using a sliding win-
dow, a target is declared after each modulation period if
and only if there is a detection in each of 3 consecutive
modulation periods. Let P3 denote the probability of one or
more false alarms occurring in each modulation period after
the 3 of 3 post detection integration for the entire set of
range-doppler cells after range and velocity unfolding. As-
sume there are n'm decisions made by the target report gen-
erator in a false alarm time T'fa* Following Marcum's
definition, let the probability equal one-half that one or
more false alarms will occur in n' opportunities. As in
Eqn. (2.5-4), it follows that
0.5 = I - (I - P) m (2.5-10)
With reference to Eqn. (2.5-6), let t represent the frame
time, TF, and na denote the number of false alarms allowed
in one frame time. For the baseline radar, t = 10 seconds
and na = 2. Hence, the false alarm time is obtained as
59
Tf a- . 5 seconds.
From Eqn. (2.5-9), Marcum's false alarm time is
Trf. - (0.69)(5) - 3.45 seconds.
Even though the post detection integrator requires 3
modulation periods to make a decision, a sliding window is
used such that decisions are made every modulation period
for each range-doppler cell. Hence, the number of indepen-
dent false alarm opportunities per second, o, equals the re-
ciprocal of Tm, the time duration for a modulation period.
This results in
n', - aTfr. 3.45
Let nf be the number of doppler filter cells used to de-
tect targets having specific doppler shifts irrespective of
range. Also, before post detection binary integration, let
pif denote the probability of having one or more false
alarms for each doppler filter arising in the ith modulation
period where i = 1, 2, 3. Finally, let Pr be the probabil-
ity that the resolved range falls within the target report
range. The probability Pf3 of generating one or more false
60
alarms in a particular doppler filter after 3 of 3 post de-
tection integration is then given by
Pfs ' PH P2f PMf P r (2.5-11)
With respect to any 3 consecutive modulation periods, one
or more false alarms result if one or more false alarms oc-
cur in one or more of the nf doppler filters. As a conse-
quence, the probability of one or more false alarms in a
modulation period after post detection integration for the
entire set of range-doppler cells involved is given by
P3 - I - (1 - Pf3)"f (2.5-12)
Let the average number of uneclipsed range gates in a PRI
be denoted by nr. To allow for variability in the doppler
shift due either to target motion, system instability, or
noise perturbations, the threshold crossing in the (i-l)th
modulation period for a doppler cell centered at frequency
f. is correlated in the ith modulation period with threshold
crossings from doppler cells positioned within the frequencyinterval, (f- Wfi/2, fj + Wfi/2 ) where the doppler filter
61
width is taken as unity and Wfj is a multiplicative constant
greater than unity. Let pf denote the single cell false
alarm probability before post detection integration of a
particular range-doppler cell. The pif's in Eqn. (2.5-11)
for i = 1, 2, 3, are then given by
P -M1 - Pf) nr (2.5-13)
P2 f ' -( - Pr'rW (2.5-14)
PSf M 1 0 ( - Pf) nrf (2.5-15)
Substitution of Eqn. (2.5-11) for P f3 and Eqns. (2.5-13)
through (2.5-15) for pif's into Eqn. (2.5-12) yields
P3 1 1- 1-[l-(I-p f) rJ11 (I-pf)flrWfZ~fj..(J..pf)frWfSJ p nf (2.5-16)
Eqn. (2.5-16) can be simplified by use of the approximation,
(IX)k -- 1-kx , for xcdl
Noting that the false alarm probabilities P f3 and Pf are
much less than unity, it follows that
P3 t- flf nr f W 2 Wf 3 Pr Pf (2.5-17)
62
Since P3 is much less than unity as well, Eqn. (2.5-10)
also simplifies to
0.5- n7m P3
from which P3 is obtained as
0.5P3 I-F .(2.5-18)
Substitution of P3 from Eqn. (2.5-18) into Eqn. (2.5-17)
and solving for pf yields
0.5 1/3 (2.5-19)
Eqn. (2.5-19) is the desired result. Given the system false
alarm specification, false alarm time is readily determined
from which the number of opportunities in terms of number of
modulation periods in that false alarm time can be deter-
mined. With n'm so determined from the system false alarm
specification and given the system parameters, nr, nf, Wf 2 ,
Wf3 , and Pr' the cell false alarm probability, pf, is
readily determined.
An alternate expression for the single cell false alarm
probability pf before M of N post detection integration is
63
now developed. The single cell false alarm probability af-
ter M of N post detection integration is related to pf by
N! 1 N-(Pfa M " i!(N-i)! Pf ) i (2.520)
The development begins by assuming a simple, continuously
operating radar where a decision is made for each range gate
sample. The resulting expression for Pfa' obtained from
Eqn. (2.5-8), is then modified to take into account the mode
of operation of a more complicated pulse doppler radar. This
enables introduction of system factors which are useful in
radar design. The resulting expression for pf from these
system parameters is shown to be consistent with Eqn.
(2.5-19) which was developed using Marcum's approach.
Consider a radar operating continuously at a constant PRF
where a decision is made for each range gate sample. Let
where the radar pulse has width r and bandwidth B. The number of range
gates in a false alarm time Tf. is given by
T- mBT .
The range gates and pulse repetition intervals within the
false alarm time are illustrated in Figure 2.5-1a. Note
64
iI I Irange ate width I __I
I-- G -'-- t im e
(a) range gates and PRI's in a false alarm time, TEa, in acontinuously running radar
ith range cell from each of N1 PRI's
- G range-doppler cell
doppler filterNGtotal number of range gates in a PRI
(b) mapping of range gate samples onto range gate-doppler filtercells
-I r 77 1///////////
1-3-
rB
doppler filter
(c) time, range gate, and doppler filter utilization in a highPRF radar
Figure 2.5-1 Process by which False Alarms Occ ur
65
ambiguous range true range
I PRI i RC Rcmslant 11
slant 2 1PIriI ~
(d) range unfolding
(mi-r. )Wn o el
associate'with nr range cells in (i-l)thfj slant over Wf(i.l)x doppler cell width
(3; rB
T -.4 associate with nr range cells in Mthfj slant over WfM x doppler cell width
(e) actual M of N correlation process (after filter unfoldingand normalization)
- - one modulation period
sliding windowsM slants @ one slant step
(f) sliding window
Figure 2.5-1 Procecas by which False Alarms Occur (cont'd)
65a
that the average number of false alarm opportunities per
second is
a =B.
Consequently, Eqn. (2.5-8) becomes
,a - 1 (2.5-21)
where the denominator equals the number of false alarm op-
portunities in a false alarm time.
Next, consider a pulse doppler radar where NI samples for
each range gate are processed through an FFT. Note that
consecutive samples are separated by a PRI. Corresponding to
the NI DFT coefficients, NI doppler filters are created for
each range gate. Thus, the number of false alarm opportuni-
ties after the FFT operation remains the same as before. The
total number of range gate samples before the FFT operation
is transformed into an equal number of range gate-doppler
filter cells. These are illustrated in Figure 2.5-lb for
one modulation period.
• mmmm mmma m m~ 66
In M of N binary integration the FFT samples from N con-
secutive modulation periods are jointly examined for M or
more coincidence detections for each of the range-doppler
cells. When decisions are made for every N nonoverlapping
modulation periods, the number of false alarm opportunities
in a false alarm time is reduced by this factor. As a re-
sult, Eqn. (2.5-21) is modified to read
N (2.5-22)Pfa = B Tf(.2
In addition, for a modern pulse doppler radar, the number
of false alarm opportunities in a false alarm time must be
modified to take into account time overhead, range eclips-
ing, and doppler filter blanking. The amount of time utili-
zation, range gate utilization, and doppler cell utilization
are illustrated in Figure 2.5-1c. Furthermore, the
multiplier effect of range and velocity ambiguity resolu-
tion, and the coincidence detection on a sliding window ba-
sis must be included. These factors are considered below.
Time Utilization Factor, KI:
As explained in Section 2.1, because of time overhead, only
a porti- of the modulation period generates range gate
67
samples which are used for signal integration. The time uti-
lization factor, denoted by K,, is the fraction of a modula-
tion period actually used for an FFT. It is given by
(NI)(T,)K 1 =Tm
where Tm and T are the time duration of a modulation period
m r
and the average time duration of a pulse repetition inter-
val, respectively. Thus, NITr is the average time duration
of a CPI.
Range Gate Utilization Factor, K2:
In practice, to protect against burnout, the receiver is
shut down during pulse transmission and shortly thereafter.
Therefore, a few range cells at the beginning of each PRI
are eclipsed and do not enter into further processing. Let
K2 be the fraction of range cells which are not eclipsed.
Denoting the total number of range cells in a PRI as NG, the
number of eclipsed range cells as rB, and the average number
of uneclipsed range cells as nr, K2 is given as
K NG-rB nr2 =N
Velocity Ambiguity Factor, K3 :
To avoid velocity ambiguities the PRF should be greater
68
than or equal to twice the doppler frequency corresponding
to the maximum possible target radial velocity. The radar
under examination, due to design considerations, uses a PRF
only half as large. Consequently, there is an ambiguity as
to whether an observed doppler frequency corresponds to an
opening or closing velocity. Use of different PRF's enables
resolution of this ambiguity. Hence, the number of resolv-
able doppler frequencies is actually 2NI. To account for
this effect, the multiplier
K S = 2
is introduced.
Doppler Filter Utilization Factor, K4:
A number of doppler filter cells located around the
mainbeam clutter doppler center frequency contain residue of
returns from mainbeam clutter and undesirable slow moving
ground targets. Therefore, these filters are blanked to pre-
vent detection resulting from these undesirable targets and
clutter. The fraction of unblanked doppler filters, K 4' is
given by
nf/2K4 = NI
where nf is the total number of unblanked doppler filters
69
after doppler filter unfolding and NI, the dimension of the
FFT, is the number of filter cells before correcting for the
velocity ambiguity factor, K3.
Range Correlation Factor for Range Ambiguity Resolution, K5 :
The concept of resolving range ambiguity arising from a
high PRF waveform was introduced in Section 1.1.1. A set of
M slants with distinct PRF's, each having mi number of range
cells in the respective PRI, is used in a beam dwell. The
Chinese remainder theorem allows the resolution of the range
ambiguity when the specific ambiguous range cell number in
each slant is correlated over M slants. The Chinese remain-
der theorem requires that the mi's are relatively prime num-
bers. The range ambiguity resolution process is referred to
as range unfolding and is illustrated in Figure 2.5-1d.
The unambiguously resolved maximum range span, RcM , mea-
sured in a range cell unit is
MRcM - ml1 2 -- mM = H mii-i
Recall that NG denotes the number of range cells in a PRI. It
follows that
NG = M =M
70
Under the condition,
Ni-l << 1; i*j, i, j=
the mi's are approximately equal. RcM can then be ap-
proximated by
RCM = (NG)M = (rB+nr)M
where nr is the number of uneclipsed range cells in a PRI.
These nr number of range cells in a particular doppler fil-
ter are correlated with nr number of range cells in the cor-
responding doppler filter in the other M-1 slants. As ex-
plained before, the doppler filter width for the ith slant
for correlation with the (i-l)th slant is widened by a fac-
tor, Wfi. Thus, the number of possible M-tuples in M slants
for correlation for a particular filter is
(ml-rB)(m 2-rB)Wf 2--- (mM-rB)WfM = 2 Wf2 -
Since there are nf number of doppler filters the number of
possible M-tuples in M slants for all filters becomes
f M NMInf n j=2 Wfi"
This process is shown in Figure 2.5-le.
71
In particular, mi and M are chosen so that RcM is much
larger than RcT, the maximum reported target range, measured
in a range cell unit. Because detections in unambiguous
range cells beyond RcT are not reported, false alarm oppor-
tunities corresponding to these cells should be excluded
from the total count. Assuming the false reports after
correlation of M-slants occur with equal probability in any
unfolded range cell, the false alarm opportunities are ef-
fectively reduced by the factor equal to the probability Pr
that the false reports fall within the reported range:
RcT ReT
(nr+rB)
Let K5 be defined as the ratio of the number of false
alarm opportunities due to range unfolding to that where
there is no range unfolding. Since there are exactly (nfnr)
opportunities without range unfolding, K5 is given by
MKs =p (2.5-23)
Sliding Window Factor, K6:
The factor of N was introduced in Eqn. (2.5-22) on the
assumption that a decision is made once every N
nonoverlapping modulation periods. When a sliding window is
72
used, a decision is made once every modulation period by
utilizing the previous (N-1) modulation periods along with
the current modulation period as shown in Figure 2.5-if.
Consequently, the number of false alarm opportunities in-
creases by a factor of N. To account for this effect, the
multiplier
I 1, no sliding windowK6 =
N, with sliding window
is introduced.
Introduction of correction factors, K1 through K6 , asso-
ciated with practical radars into Eqn. (2.5-22) represent-
ing an ideal radar results in
N 6 -1Pfa =-, I__ I~ [. (2.5-24)= BTf-- - i=,
It is now shown that Eqn. (2.5-24) leads to an expression
for the single cell false alarm probability pf which is con-
sistent with Eqn. (2.5-19). Recall that Eqn. (2.5-19) was
obtained on the basis of 3 of 3 post detection integration
using Marcum's method. Letting M=N=3 in Eqn. (2.5-20), it
follows that
(- 3Pf a = P7
73
Hence, Eqn. (2.5-24) can be rewritten as
SN (2.5-25)Pfa = P = BTf. K1K2K3K4KSK6 '
Substituting M=3 into Eqn. (2.5-23) gives K5 as
n=nrWf 2 Wf3KS= nrnf T -
Therefore, the factors which enter into Eqn. (2.5-25) are
N=3, K, K2 = 3s2, /2
n3 Wn f n r W f 2 W f 3 l O 3Ks = n1 nr Ke=3.
Hence, Eqn. (2.5-25) is given by
3 3Pf = ( fa(NIT (nr () ( ,/-2 N wf2wf3Pr (3)
1TM NGII nr
With the sliding window, there is one false alarm opportunity
per modulation period. Thus, there are
nm= Im Tf a
_Tta
false alarm opportunities in a false alarm time. Substitu-
tion of
Tf.atnmTm, B _ I/r, and NG T"/T
74
into Eqn. (2.5-27) gives
Pf = 3 1 (2.5-28)fnmnfnr Wf2Wf3Pr
or, equivalently,
Pf =''r nrrtW; 2We3Pr] 1/3 (2.5-29)
Since
T;. - 0. 69Tf.
it follows that
nm = O. 69n m
Therefore, Eqn. (2.5-19) can be rewritten as
1r 0.5/0.69 1/3 (2.5-30)Pf =-nrL nmnfWf 2Wf 3Pr
=4r[ 0. 725 1 /3tn mrflWf 2Wf3Pr ./3
Note that Eqn. (2.5-30) and Eqn. (2.5-29) are consistent in
the sense that they yield approximately the same values for
Pf. It can be concluded, therefore, that the two different
approaches by Marcum and Barton produce equivalent results.
For M of N post detection integration, substitution of
75
Eqn. (2.5-20) into Eqn. (2.5-24) yields
N N! N-i NSi=M i!(N-i)! Pf ) BTf K1K2KK 4K 5K 6 (2.5-31)
From a system designer's point of view, this equation is
convenient for evaluating pf, the cell false alarm probabil-
ity before M of N post detection integration. The results
obtained using Eqn. (2.5-31) are summarized in Table 2.5-1
for the 3 of 3 3-slant baseline configuration, a 2 of 3
3-slant configuration, and a 2 of 2 2-slant configuration.
Relative to the 2 of 3 3-slant configuration, the value of
Pf is maintained the same as that determined for the
baseline configuration. This is equivalent to holding the
threshold fixed for both configurations. As a result, the 2
of 3 configuration is able to achieve more detections at the
expense of a higher false alarm rate. This is indicated by
the larger value for Pfa and the smaller value for Tfa*
With respect to the 2 of 2 configuration, the false alarm
time is maintained at the same value found for the 2 of 3
configuration. This allows comparison of detection perfor-
mance for the two configurations on the basis of having the
same false alarm performance. The comparison is presented
in Chapter 4 where it is used in conjunction with
scan-to-szan processing.
76
As a check, it is now shown that the numerical value de-
termined for pf for the baseline 3 of 3 post detection in-
tegration is consistent with the baseline system requirement
that there be on the average 2 false alarms per scan. Given
that pf=7.6xl0 " , it follows that
Pfa , Pi = 4.39x10"1 •
The total number of false alarm opportunities per scan is
given byP TF
nrWf2 Wf 3nfP( rim"s-iE )
- (42)0(2.2)(2)(1 95JA!.73 40)
= 4.48xl0p .
The average number of false alarms per scan equals the prod-
uct of P fa and the total number of false alarm opportunities
per scan. This yields
(4.39x10-%)(4.48x]09) = 2
which is the expected result.
77
Table 2.5-1 False Alarm Probability for the ProcessingOptions Evaluated
3-slant 2-slant
M of N 3 of 3 2 of 3 2 of 2
time normal reduced normal reduced normal reducedoverhead
FA/scan 2 2 4.15x10 3 4.60xl10 3 4.15xl10 3 4.60x10 3
N fMP's/ 1040 1040 1040 1040 6993 693
scan)
T fa (sec) 5 5 2.41xl0 3 2.17x103 2.41xl0 3 2.17x103
B(MHz) 1.25 1.25 1.25 1.25 1.25 1.25
K 1 0.479 0.652 0.479 0.652 0.653 0.768
K 2 0.933 0.933 0.933 0.933 0.933 0.933
K3 2 2 2 2 2 2
K4 0.762 0.762 0.762 0.762 0.762 0.762
K 5 533 533 282 282 282 282
K 6 3 3 3 3 2 2
P fa X10-- 4.39x10 a- 3.23 x100 a- .73x1 4 1.41x104 Wl .27x104 Zl.20x4
Pf'1O 4 7.60x10 0 6.86x10 0 7.60x10 0 6.86x10 0 l.13xl10 1 1.09x10 1
Notes:
1. For computing K 5 the following parameters are used:
n -42, r -3, R cT 61 73 , W f1 =2.2, W f2=2.03 .
2. Explanations for the numbers under columns labeled 2 of 3 and 2of 2 with normal time overhead are given in Section 4.3. Thenumbers with reduced time overhead can be found similarly.However, it should be noted that the number of range gatesamples no longer matches with the total number ofrange-doppler cells. It is because zero filling is requiredto generate FT samples when the input time samples in a CPIare not equal to integer powers of 2.
78
2.6 Target Models Assumed
Before presenting the target models used in the analysis
of detection performance, the concept of target radar cross
section (RCS) is reviewed. Target RCS is related to the ra-
tio of received power from a target to the power incident
on the target. The received power is expressed in terms of
the cross sectional area of an isotropic scatterer which,
for a given incident power, would produce the same received
powe as the actual target. From this concept, the radar
cross section, 0., is given by
a = 4r power reflected toward source/unit solid angle
incident power density
=47rR2 I. 12
where the parameters are
R: range to the target, assumed to be large enough suchthat the target is in the far field from the transmit-ting antenna
E r: reflected field strength at the radar
Ei: incident field strength at target.
It follows that the RCS of an isotropic reflector, e.g., a
sphere, is equal to its projected area normal to the direc-
tion of radar illumination. Most targets are not isotropic
79
and exhibit directional preference. Objects with the same
physical projected area can have considerably different val-
ues for RCS. For example, at S-band, the RCS at nose-on as-
pect of a cone-sphere is 30 dB smaller than the RCS of a
sphere with the same projected area while a corner reflector
can have an RCS 30 dB greater than that of the sphere, all
having the same projected area of 1 m2.
Skolnik (9] and Nathanson [10] each have a good summary
of work by various investigators on target RCS and commonly
used statistical target models.
The RCS of a large complex target may be approximated by
assuming that the target is composed of individual reflec-
tors such that the total value of a is related to the vector
sum of the individual cross sections, k' in the follov.ing
manner:
a= I EZ-ae4 _{',) 1 2
where dk is the distance from the radar to the kth reflector
with RCS, Tk"
The RCS of a complex target is a strong function of the
aspect angle. Since the precise aspect angle is unknown in a
80
given situation, the RCS is best described statistically. A
summary of the frequently used target models is presented
below:
1. Marcum case (nonfluctuating target):
For nonfluctuating targets, Ois a deterministic constant.
Let A denote the envelope amplitude of the received signal
voltage. Since the average power during the duration of the
pulse, A2/2, is proportional to a, it follows that
K a = A 2
where Ka is the constant of proportionality. A constant
value of a implies a constant value of A.
2. Swerlina case 1:
This model assumes that a target is composed of many
reflectors where none is dominant. For this case, c remains
constant over all N slants within a scan. However, - varies
randomly from scan to scan. The probability density func-
tion of the RCS is given by
{exp >i, 0p(a) (2.6-1)
c<O
where av denotes the expected value of a To represent the
81
variation of signal-to-noise ratio from scan to scan due to
a target's RCS fluctuation, it is convenient to have an ex-
pression for the probability density function of the inte-
grated signal-to-noise ratio, (S/N). It is the ratio of the
received time averaged signal power to the mean noise power
during the duration of the received pulse train integrated
in a slant. Let S0 denote the received signal power assuming
no beam shape loss. Since the mean noise power is assumed to
be constant, (So/N) is directly proportional to RCS. It
follows that
(J Kba. (2.6-2)
Averaging over the fluctuations from scan to scan, the aver-
age signal-to-noise ratio is
- = Kb'7& . (2.6-3)
Therefore, Eqn. (2.6-1) can be transformed according to
P(so) = P(a) do a
N S
( exp , S 0/N>O(SO/N) So/N
= (2.6-4)0, So/N <.
82
The probability density function for the envelope ampli-
tude can be similarly derived using the transformation,
K~a = A2
from which it is found that
-- 2A A 2 (2.6-5)
Then, p(A) becomes
a)I do -LA exp -A) ,a A 2 .,/K , 2 2
p(A) - (2.6-6)
0, A<4.
3. Swerling case 2:
This target model is assumed to be composed of many re-
flectors where none is dominant as with Swerling case 1.
Therefore, the probability density function for 0 is identi-
cal to that in Swerling case 1. However, now the RCS is as-
sumed to fluctuate from slant to slant while the amplitude
of the received pulses within a CPI remains constant. This
rapid fluctuation is not encountered with practical targets.
However, it can be induced by slant-to-slant frequency agil-
ity.
83
4. Swerlina case 3:
This target model is assumed to be composed of many equal
size scatterers plus one dominant reflector which fluctuates
slowly from scan-to-scan. As with Swerling case 1, a remains
constant over all N slants within a scan. The probability
density function of this model's target RCS is given by
4a, ex - 0 2
p(a) = 1(2.6-7)
where, as before, rav denotes the expected value of c. The
probability density function for the integrated
signal-to-noise ratio , p(S0/N), is obtained from Eqn.
(2.6-7) with the transformation given by Eqn. (2.6-3). Mak-
ing the substitution, Idr/d(So/N)=l/Kb, into Eqn. (2.6-3)
gives
4(So/N) C -2(So/N) S IS/N O
(So/N) 2 ('o/ N)PL () = / (2.6-8)
The probability density function for the envelope
amplitude A is again obtained from Eqn. (2.6-7) with the
84
variable transformation
Kao = A2, and Kcv =A. (2.6-9)
Use of , Idr/dAI, and or, given by Eqn (2.6-5), yields
SA 3 &xi( A-)
p(A) 0(2.6-10)0O, A<O.
5. Swerling case 4:
Swerling case 4 has the same probability density function
for V as Swerling case 3. As in case 2, the RCS is assumed
to fluctuate from one slant to the next while the amplitude
of the pulses in a CPI remains constant. Swerling case 4 is
related to Swerling case 3 as Swerling case 2 is related to
Swerling case 1.
These probability density functions are used for deter-
mining detection probabilities in the baseline and modified
radar configurations described in the Chapters to follow.
85
CHAPTER 3
PERFORMANCE OF THE AIRBORNE SURVEILLANCE RADAR IN ITS
BASELINE CONFIGURATION
Detection probabilities in the baseline configuration for the five target
models introduced in Chapter 2 are determined in this chapter. This is done by
taking into account the number of pulses available for integration in each
slant, the beam shape loss, and the cell false alarm probability also estab-
lished in Chapter 2.
3.1 Sufficient Statistic and the Likelihood Ratio Test (LRT)
With reference to the receiver/processor block diagram of Figure 1.1-2, it is
assumed that the analog-to-digital converter (ADC) and the clutter canceller are
both ideal with their characteristics as shown in Figure 3.1-1 (a) and (b), re-
spectively.
output magnitude
input -r0tfmnF
frequency
(a) ADC response (b) clutter canceller response
Figure 3. 1-1 Assumed ADC and clutter Canceller Characteristics
86
Let v(t) denote the received waveform at the IF filter output. Then, v(t) is
given by
n(t) Ho
V(t) = (3. 1-1)s(t-t')p(t-t )+n(t) HI
where Ho and H, denote the null hypothesis (target absent) and the alternate
hypothesis (target present), respectively. Under Ho, v(t) consists of the noise
n(t) alone. The rectangular gating function p(t) in the expression of v(t) un-
der H, is of unit amplitude and duration r. The clutter c(t) is not included
in Eqn. (3.1-1) since the clutter spectrum is assumed to be confined in the stop
band of the clutter canceller. Therefore, the detection problem under consider-
ation is for target signals with doppler frequencies greater than fdrin
which are .embedded in thermal noise. The delayed signal s(t-ta) can be
written as
s(t-t ) = Acos[(wMF+d)(t-t ) + 0( t-t'd) + 0] (3.1-2)
where A is the signal amplitude, 6(t) is the signal phase modulation, 0 is the
unknown initial phase, and t' is the round trip delay for a point target
located at range R. The time delay is given by
2Rtd c
where c is the speed of light.
Corresponding to the received signal with delay t', there is a time gat-
ing pulse p(t-td) which effectively multiplies the incoming signal. In general,
87
t' is not equal to td. The mismatch between td and td gives rise to
a range gate straddling loss, the average value of which is included in the sys-
tem loss factor. Having accounted for the straddling loss in this manner, it
is assumed in the subsequent analysis that td=td. Since the integrator
following the multiplier (mixer) is synchronized to td and performs the inte-
gration over the duration of p(t-td), p(t-td) equals unity during the entire
integration period and can be dropped without effect. For the radar under consid-
eration, there is no phase modulation. Hence, 0(t) = 0. The radar receiver un-
der consideration implements the Bayes strategy by performing the likelihood
ratio test (LRT) in each range, angle, and doppler frequency resolution
cell. Thus, without loss in generality, the delay in the expression for s(t) can
be dropped. Then, v(t) becomes
n(t) : under Hov~)=; Qct_.<r. (3.1-3)
s(t)+n(t) : under H,
The expression for s(t) can be written in terms of its quadrature compo-
nents s(t) and sQ(t) as
s(t) = Acos[(M'F+Wd)t+O]
= Re( ((t) eU wI~t+O)]}
= Re { [s(t)+jsq(t) ] expLKjwFt+O)])
= s( t)Cos(IFt+O) - sq(t)sin(ujIFt4#) (3.1-4)
where
S = A eJldt = si(t) + jsq(t) - Acostwdt + jAsinwdt
88
is the complex envelope of s(t). The narrowband noise process n(t) can
also be expressed by its quadrature components as
n(t) = nl(t)cOSwlFt - nq(t)sinFt. (3.1-5)
The receiver that maximizes the output signal-to-noise ratio is the correlation
receiver as shown in Figure 3.1-2. This also maximizes the probability of detec-
tion when v(t) is Gaussian distributed.
N. 4 Ast N(FFT
yE Q(e, mq(kj)
2 Asin lFt Xk
v(t). E 4(Reo},2Im,Mo) r >
Y,(-r, YlXI(k)
tl")dt . N -FFT
2
ti
2 Acos4Ft
Figure 3. 1-2 Equivalent Receiver block diagram
89
In Figure 3. 1-2, the constant 2A in the reference signal going into the product
mixer is chosen such that the variancesof yI(r, t1) and yQ(', ti) ate unity where No/2
is the two sided noise power spectral density and N is the single pulse
peak signal-to-noise ratio. The noise power contained in a video bandwidth B is
equal to NOB. Thus, the single pulse signal-to-noise ratio (S/N) at the input
to the FFT before correction for the loss factor is
(- a(l1/2)A ( 1/ 2 316N NOB No T
The multiplier l/,4"Nfx before the FFT in Figure 3.1-2 is introduced so that
the variances of Xx(k) and XQ(k), appearing at the FFT outputs, are each unity.
Returning to Figure 3.1-2, the output of the product mixer is described by
first considering the signal and noise separately. Let the constant K represent
K= 2A
In each of the I and Q channels, repectively, the output of the product mixer for the
signal is
Ks(t)cos(tiFt) = K(sI(t) -[cosO+cos(2uzpt+O)]
-Sq(t) -[sinO+sin(2wxFt-0)] (3. 1-7)
and
90
Ks(t)sin(wL.t) - K(s,(t) I -sinObsin(2uvt+O)]
-S'Q(t) -2[coO cos(2ukLgt4O)]} (3. 1-8)
Similarly, the output of the product mixer for the noise is
Kn(t)COs( Ft) =K( -2n1( t)[1+cos( 21a4Ft) ] - -fnQ(t)sin(24Ft)) (3.1-9)
and
I IKn(t)sin(MFt)-K(- -2nQ(t)[l-cos( 2 IFt)J + -2nI(t)sin(2 jFt)) (3.1-10)
The sampled output of the integrator is obtained by recognizing that the double
frequency terms integrate to zero while the narrowband signal and noise do not ap'-
preciably change during the integration period r. Thus, the sampled output of the
integrator is closely approximated by
Kfti' s(t)cos(at)dt = Kr { sI(tl)cosO-sq(tl)sinO)
-- Ar ( COS(Wdti)cosO-sin(wtl)sinO4
K=-tA cos(wdt1 ) (3.1-li)
and
91
Kf' s(t)sin(UlFt) dt = -- , (si(t)sinO+s(t)cosO)
1K
-K Ar { cos(wdtl)sinO+sin(wdtl)COSO)
-K Ar sin(wdt/,$) (3.1-12)
The sampled output of the integrator for noise is
tI+d KrKf t n(t)cos(MFt)dt - n,(tl) (3.1-13)
and
K " I n(t)sin(UtFt)dt - -I- nQ(t/) (3.1-14)
Thus, the sampled output of the integrator is
K rn,(tl) . under Ho
yi(r, t/) -(3.1-15)
Ar cos(wdt/41)KI rnl(tj) under H1
and
- K2 rr,(tl) under H0yQ (r, td) = (3.1-16)
~ 1) K K-- Ar sin(wdt,4+)- - rnQ(tl) under H(
92
The mean and variance of y, and yQ are
Yi(r, t/) :
under Ho:
mean = 0
(2Ar 2variance 2NOA{-J n,(ti) 1
under H1:
29kmean r COS(Wdt/+4 ) COS(W~dtj4 )mean = 2N----t,-I = o~aht 4
2NO -N-
variance = I
YQ(r, tl)
under Ho:
mean = 0
variance = I
under Hi:
mean = -4F sin(wdt+4 )
variance = I .
As shown in Figure 3. 1-2, let
XIMl =-L yI( r , ti)
and
XQ - y(r, t)-
93
It is convenient to consider the FFT process for noise and signal separately.
Let n([) and n (I) denote the eh in-phase and quadrature noise sample in a
sequence for 1 - 0, 1, - -, NI-1
nl(1) - nl( lTrr )
and
n (/) = Kr," I2 L[N'In(Ir
The kth output coefficient Ni(k) of the NI-point FFT in response to the input
sequence n'(1) is given by
N1-1 2WI
NI(k) - E n'(1) e" N, k = 0, 1,--,NI-I. (3.1-17)1=0 1(.-7
Note that nj(l) is a zero mean white Gaussian noise sequence. Hence,
E[nj(/ = 0 and E[nI(l)n,(m) ] =- 1 "
The first and second moments of NI(k) are given by
S2zk1NI-1 _j -"k1E[NI(k) E[n,(l)] eNI = 0 (3.1-18)
and
NI-1 NI-1 2wkl .2 pm
E[N I(k)N(p)] = E E[n'(/)n'(m)] e" - - (3.1-19)
94
respectively. Note that the variance of n'(1) is .2 = N' . Letting k=p, the
variance of Ni(k) is given by
N1-1 N1-1 _j 2xk(l-m)NE N e' N1var [Ni(k)]= E I=8-N1 e N
NI-1= N',=1.
For k*p, Eqn. (3.1-19) becomes
Iep 27rk-p) ]NE[NI(k)No(P)] = I -exp _ 27r(k-p)
= 0 for k'p
Identical results apply to the quadrature component NQ(k). Hence, the noise outputs
in different doppler cells are orthogonal. Because they are also zero mean, they
are uncorrelated. NI(k) and NQ(k) are also Gaussian random variables because they
are each linear combinations of jointly Gaussian random variables. Therefore, N1(k)
and N(p) as well as Nq(k) and Nq(p) are independent for klp.
The expected value of the cross term, E[N,(k)N;(p)], is given by
N-2*k- .2pm'1 NI-1 -J'j +J- -E[N1 (k)NE(p)] = E E[nj(l)n4(m)] e eE[NI~) N; =0 m=O
A property of a stationary narrowband noise process of bandwidth B is that
the crossspectral density SNINQ(f) of its quadrature components is purely imaginary
and is given by [13]
95
SNJNq(f) - -SNQN(f)
j[SN(f+fo)-SN(f-fo)] -B < f <B
0 ; elsewhere
If the bandpass noise n'(t) is Gaussian with zero mean, and its power spectral
density SN(f) is locally symmetrical about the midband frequency _+fo, then' SNjNQ(t)
equals zero. It follows that the in-phase noise n (t) and the quadrature noise n4(t) are
orthogonal, i.e., E[n(0n4(m)] = 0 for all 1, m. Therefore, NI(k) and NQ(p) are
also orthogonal. Since both are zero mean Gaussian distributed, they are statistically
independent.
Next, the kth output coefficient S(k) of a DFT in response to the input
signal sequence s (/)+js4(1), I = 0, 1, - -, N-1, is considered where I denotes the
signal sample taken at t = Tr - i/Ft.. From Eqns. (3.1-15) and (3.1-16) and Figure
3. 1-2, observe that
Sj(l) = K Ar cos(21lrfd/Fr+)
s4(1) - K Ar sin(27rlfd/F, 4)
Let s'(1) denote the weighted sum
s'() =
= K Ar e j ej2 dfd/Fr
Sd ej e j2 wfd/Fr (3.1-20)
96
where
d -KAr.
Tne DFT output SMk coresponding to the input sequence s'(1), 1 =0, 1, --
N1-l, is
S(k) E (E jOe -l=0
= 0e 1 - ei2xfd/Fr-k/N)NI1 -J e j2xfd/Fr-k/Ni)
-d ejo eji(fd/Frrk/N)(I-1) 5f(d/r4CN)I(.-1sinl(fd/Fr-k/Nl) (.-1
Let a and denote
o:- 7ir~f/ k/N 1 )
= 7r"fd/Fr+ k/N1 )
Note that the DFT outputs, SI(k) and SQ(k) corresponding to input sequences,
si(l and sV(), respectively, are
-[ jecN1 snct+ eC'eAI1 siC, (3.1-22)
2sinNa sinN1l
SQ(k) = -4 [e-~' r- - Ce-e'ANI- 1I (3.1-23)
Clearly, the sum, S1(k) + JS() is equal to S(k).
97
When fd/F, - k/Ni, a - 0, N1,6 m 2xk, and the DFT becomes the
matched filter for the corresponding doppler frequency. Under this condition, the
DFT outputs become
S(k)fd/Fk/N d N, ej o (3.1-24)
d N I ej#SI(k)fd/Fr=k/N -TN e
d N=--N (cosO + jsino) (3.1-25a)
S ~ d NI1 ejSQ(k)I fd/Fr=k/N 2j
d N1 (cos4 + jsino) . (3.1-25b)-2j
With reference to Figure 3.1-2, the output X(k) of the summer following the
FFT's is given by
X(k) = X1(k) + jXQ(k)
INI(k)+jNQ(k) = N(k) : under H(- (3. 1-26)
Si(k)+IQ(k) + NI(k)+jNQ(k) = S(k)+N(k) : under H,
For the remainder of the discussion, it is assumed that fd/Fr = kN. The real and imaginary
parts of S(k) are determined from
S(k) = SI(k) + JSQ(k)
= Re(S,(k))-Im(Sq(k)) +jj Im(SI(k)+Re(SQ(k))]
- d N, cosO + jid N, sin.
Hence, the real and imaginary parts of X(k) are
98
Re(N1(k) - Im(NQ(k)) :under H0
Re(X~k) d N, cosq$ + Re((N1(k)) - Jm(NQ(k)) :under H, 312a
and
(lm((N1 (k)) + Re(NQ(k)) :under H
lm(X~k)) d N, sin4' + Im((N1 (k)) + Re(NQ(k)) :under H,.(312b
It will now be shown that X(k) under H., which is given by
X(k) -N(k) - Re(N(k)) + jlm(N(k))
is a zero mean complex Gaussian random variable. To prove this, the following
must be verified-
1) Re(N(k)) is a zero mean Gaussian random variable
2) Im(N( k)) is a zero mean Gaussian random variable
3) Var[Re(N(k))] = Var4Im(N(k))]
4) E[Re(N(k))Im(N(k))] - 0.
From Ecjn.(3.l-27), Re(N(k)) and Im(N(k)) are given by
Re(N(k)) - Re(N,(k)) - Im(NQ(k))
N1 -1 2,x ) id 21Z~ L n'(1)co N1' + (3.)si 1-28a)
and
Im(N(k)) - Im(NI(k)) + Re(NQ(k))
= E n'(I)sin-7r + n4(oc~ . 3.I-8b
99
Since n1(l), n'(1) are both zero mean jointly Gaussian random variables, Re(N(k)) and
Im(N(k)), which are linear combinations of n!4(l), n4(I), are zero mean Gaussian random
variables. In particular,
IE[Re(N(k))] = E[Im(N(k))] - 0 .(3.1I-29a)
From Ecjn. (3. 1-28), the variance of the Re(N(k)) can be written as
NJ-i 2irkl - 2rkl] 12var[Re(N(k))] - E[ ( E ni(lOcos +~ n(l)sin N]
Since E[ni(On'(m)] - 0 for l*m and E[n1(1n'(m)] - 0 for all 1, m, var[Re(N(k))]
becomes
NI-1 92 2(2k)) NI-i si tW 21kvar(Re(N(k))] - E IE~ni (1)]Cos 2 -NJ + E~n4 2(i)] 2'sin l
-1 . (3.1l-29b)
Similarly, var[Im(N(k))] can be shown to be 1. To prove that Re(N(k)) and Im(N(k))
are orthogonal, E[Re(N(k))Im(N(k))I is carried out as follows:
E[Re(N(k))lm(N(k))]
-E( ( Re(N1 (ic)) - Im(NQ,(k)) ) Im(N(ic)) + Re(NQ,(k)1
E E Enl(l)n'(m)1 cot - sin ,-
N- 1 N- 1 27rkl 27rkmn+ E E E1(n4() cot - cot-
N1-1 N1-1 21rk/l 27ikm- E E E[ni~~ () sin - sin--
N1-1 N-1 i 21rkl 2ikm+ E E En(n4m si- cot-
1-- M= : 4IO4mJ 1 N
100
Since E[n'(l)n,(m)] = E[n4(l)n4(m)] = 0 for lI:m, E[nj(l)n4(m)j = 0 for all I and m,
and E[nQ2(/)] _ E[n4 2(1)], the above quantity evaluates to zero. Thus,
E[Re(N(k))Im(N(k))] = 0 . (3.1-29c)
It is concluded that N(k) is a zero mean complex Gaussian random variable.
With the above preliminaries, the mean and variance of the real and imaginary
parts of X(k) under both hypotheses are given as follows:
Re(X..:))
under HO:
mean = 0
variance - Iunder Hi:
mean f AJT cosO = ,-s cosO
variance = 1
Im(X(K))
under HO:
mean = 0
variance = I
under HI:
mean = 4-% sinO
variance = I .
Recall that st - N1%P = 2N,(S/N)p is the integrated peak signal-to-noise ratio.
101
Define the envelope voltage r and the phase angle v such that r and v are re-
lated to Re(X(k)) and Im(X(k)) by
r _ ,] [Re(X(k))] 2 + [Im(X(k))] 2 . (3. 1-30a)
v = tan-, Im{X(k)) (3.1-30b)Re(X(k))
It follows that
Re(X(k)) = r cos" (3.1-31a)
Im(X(k)) = r sinv , (3.1-31b)
Note that Re(X(k)) and Im(X(k)) are statistically independent due to the indepen-
dence of Re(N(k)) and Im(N(k)).
Under H., the joint probability densty of Re(X(k)) and Im(X(k)) is
p(Re(X), Im(X)) - p(Re(X)) p(Im(X))
"Tvlr exp -(Re(X) )2 +(Im(X}) )
e (3.1-32)Tr
By the transformation of variables given in Eqn. (3.1-31), the joint probability
density p(r, v) can be shown to be
p(r, v) =-r exp(--2 ) (3.1-33)
The marginal density function p(r) is obtained by integrating Eqn. (3. 1-33) over
102
, which becomes
r2
p(r) r- e 2 dv = r exp- (3.1-34)
Under hypothesis H1, the joint probability density function of Re(X(k)) and
Im(X(k)) given 0 and 9t is
p(Re(X), Im(X)JI,9t) - - r exp ( (Re(X) - 4-t CosO) 2 + (IM(X} + 4"N sin))
(3.1-35)
By the transformation of variables given in Eqn. (3.1-31), p(r, &JO, 9 can be
obtained as
r r2 + 9t _ 2r - cos(- W) (31-6
p(r, 140, 91) = T r exp[ 2 "( ] (3.1-36)
Integration of Eqn. (3.1-36) with respect to V gives the conditional marginal
density function p(rlO, 9t) as
p(rI#, 91) = re - (r 2+ 9 / 2 I r. r e r .gt cos(*+Y) dv
=re "(r=+ / 2 lo(r'-9) , (3.1-37)
where Io is the modified Bessel function of the first kind, zero order. Note
that Eqn. (3.1-37) is not a function of 0. Assuming 0 is uniformly distributed
between 0 and 2r, the conditional density function of r given 9t is
p(rl ) o p(r,OCt)dO f. JTo' r p(r1O,'t)dO
= re-(r2 +/2 i0(r-t) . (3.1-38)
103
The envelope voltage r is the sufficient statistic. Under the null hypothesis,
the probability density function for r, p(r 9§, is Rayleigh as agiven by
Eqn. (3.1-34). Under the alternate hypothesis, p(r49) is Rician as given by
Eqn. (3.1-38), for all target models assumed. The likelihood ratio test (LRT)
decides that a target is present when the test statistic r exceeds the threshold
rT. Otherwise, it decides that a target is not present
H1
r < rT •Ho
The detection probability arising from the probability density of r given t
must be properly averaged over target RCS fluctuations which is carried out in the
next Section.
':4
3.2 Detection Threshold and Probabilities of Detection
The detection threshold is determined from the cell false
alarm probability pf prior to the binary post detection in-
tegration, and the probability density function p(r) for
the test statistic r under the null hypothesis H0 . The
false alarm probability is determined in Section 2.5 from
the given system false alarm rate and the specific post de-
tection integration involved. The probability density func-
tion for the test statistic under H0 is Rayleigh as given by
Eqn. (3.1-34). Hence, the cell false alarm probability pf
is
Pr- r -2/2=exp(- ) (3.2-1)
The detection threshold rT follows from Eqn. (3.2-1) and is
rT . 2 in~ (3.2-2)f
The probability of detection Pdi in the ith slant condi-
tioned on the integrated peak signal-to-noise ratio for the
ith slant, denoted by 9i' is obtained by integrating the
probability density function p(rIRi) under hypothesis H1 .
Let (S/N)i = NI(S/N) pi denote the integrated signal-to-noise
ratio for the ith slant. Since i . 2Ni(S/N)pi'
105
Let (S0/N) = ?0/2 denote the integrated signal-to-noise ra-
tio assuming all pulses are received through a rectangular
beam at its peak gain. Taking into account, LBSi, the beam
shape loss for the ith slant, the integrated signal-to-noise
ratio for the ith slant is expressed as
The actual beam shape losses, in dB, are determined and
tabulated in Section 2.4.
The conditional density function p(rl1i) is Rician for
all Swerling and Marcum target models assumed. Thus, for a
specified value of i' Pdi is given as
Pdi f r exp(-r 2+%/2 I0(r4 .dr - Q(rT, 4) (3.2-3)
where Q(.) is Marcum's Q function [2].
For specified value of i; i=l,2,--,N, the detection
probability PId(M,N) after M of N binary post detection in-
tegration is a function of Pdl' Pd2' - - PdN' M, and N. This
is written as
106
P;(M,N) - f(pd Pd2,--,PdN, M,N); I<M<N. (3.2-4)
For an idealized rectangular antenna beam where Pdi = Pdj'
i = j, P'd(MN) is given by
Ps , N N! i N-iP(MN) = r=EM i!(N-i)! p . •
In general, Pdi * Pdj because the antenna beam is not
rectangular and LBSi * LBSJ . Then, Eqn. (3.2-4) consists of
product terms of the form
N-i1I 1 P d ij)) , M - 32-ak=1 Pdk j=1 d 4 /I(.-a
or
Nkl P i = N. (3.2-5b)
k=1 dk
The unconditioned detection probability Pd is obtained by
averaging P'dCMIN) over the random variations of (S/N)1,
(S/N)2, - - (S/N)N. This process is illustrated in Figure
3.2-1. The unconditioned detection probability is given by
P [P M7N) ] (3.2-6)
where the overbar indicates an N-dimensional expectation
10 7
ist slant Pdl P(MNianF Pdi M of N average
i over Pd =P (MN)post RCS
detection fluctuationIintegrator
I Nth slan d
Figure 3.2-1 Process for Determininv Probabilities of De-tection after M of N Post Detection Binary Integrator
1ist sla Intj d l
Sith slan Pdi M of N P3 post
detection =Pa(Pdl,Pd2,-,PdN,M,N)integratorNth slant- dN I
Figure 3.2-2 Process for Determining Probabilities of De-tection after M of N Post Detection Binary Integrator
(slant-to-slant RCS fluctation)
108
with respect to (S/N)1 , (S/N)2, - - , (S/N)N.
For scan-to-scan fluctuation, the signal-to-noise ratio
remains constant over each slant and Eqn. (3.2-6) can be
written as
Pd = [PdW4,N )] = f' PA(M,N) p(So/N) d(SWN) . (3.2-7)
For slant-to-slant fluctuations, the signal-to-noise ratio
varies from one slant to another. Assuming the variations to
be statistically independent, each factor in Eqns. (3.2-5a)
and (3.2-5b) can be averaged separately. The N-dimensional
expectation of Eqn. (3.2-6) then involves product terms of
the form
N-i i N-i11p 11(1-p )= f-P "r'A 0 _ M<N- I
k=1 dk j1 - d (i+j) k = dk j= d(i+j)
and
N
[k01 Pdk] - k]=[ dk N,
respectively. Hence, the expression for Pd corresponding to
slant-to-slant fluctuation can be written as
109
1d , 2d - dN ) • (3.2-8)
The detection process for slant-to-slant fluctuating target
cases is redrawn in Figure 3.2-2.
The probability of detection for the five target models
introduced in section 2.6 is developed below for the
baseline 3 of 3 post detection integration. For this pur-
pose, the center slant is designated with a subscript 1, and
each of the two outer slants is designated with a subscript
2.
1. Marcum (Nonfluctuating) case:
For this model, the probabilities of detection in the
center and outer slants are:
p = Q(rT4") (3.2-9a)dI
pd2 = Q(rTT4 ) (3.2-9b)
where
i = 1, 2.
110
Since this model assumes no fluctuation, there is no averag-
ing with respect to RCS fluctuation. Thus, the detection
probability after 3 of 3 post detection integration becomes
P (3,3) f Pd =p Pd 2 (3.2-10)
2. Swerling case 1:
For this model, Pdi' and P'd(3,3) are identical to the
Marcum case given above. Pd is obtained by averaging
PId(3',3) over the scan-to-scan variations of (S0/N). Hence,
Pd is given by
Pd f~ [P (3,3)jI p(S0 /N) d(S0,/N)0 0 /
" fo[Pd(3,3)] exp - N) d(So/N) (3.2-11)SWN S0/N
where (S0/N) is the mean integrated signal-to-noise ratio
corresponding to the mean target radar cross section.
Further simplification is not possible and the integrand
must be evaluated numerically.
3. Swerling case 2:
For this model, Pdi is the same as for the Marcum case.
Hence, Eqns. (3.2-9a,b) are applicable. Pdi for the ith
slant is obtained by averaging Pdi over the slant-to-slant
111
variations of (S/N)i due to target RCS fluctuation:
d= J0i[Pd ] p([S/N]i) d([S/N)i] . (3.2-12)
Substitution of Eqn. (3.2-3) for Pdi in Eqn. (3.2-12) re-
sults in
= j~o ( fr exp [-(rl+2(S/N)(LBs)/2]
•Io [r. S N)(i~7)] dr )P(S/N)d(S/N)
where p(S 0 /N) is given by Eqn. (2.6-4). Interchanging the
order of integration transforms the above to a product of a
gamma function and a confluent hypergeometric function [2].
Using the power series expansion of the confluent
hypergeometric function leads to a simple closed form solu-
tion given by
In pdi exp ( I ) ; i = 1, 2, (3.2-13)l+(S7~ql)
where (SN) i = (S 0 /N)(LBsi)" The probability of detection
Pd after post detection integration is
Pd = [PI(3,3)] = gp'PdM,N) = -d d 2 (3.2-14)
112
4. Swerling case 3:
The procedure in this case is exactly the same as that
used for Swerling case 1 except the density function for
(S0/N) corresponds to one dominant plus Rayleigh (exponen-
tial power) scatterers. As with Swerling case 1, Pdi and
Pd (3,3) are identical to the Marcum case. Pd is given by
Pd Jo [P;(3,3)] p(SWN) d(So/N)
= 0 '(34S /N (2S 0 /N d SO/N) (3.2-15)
(SoN)Z SWN
The above integral must be evaluated numerically.
5. Swerling case 4:
Exactly the same steps are used for this model -as in
Swerling case 2. However, the form of the probability den-
sity function for the variation of (So/N) is identical to
that of Swerling case 3. From DiFranco and Rubin (2], Pdi
for this case is given by
--d d] p([S/N]i) d([S/N)i]
1 1I + In I+f exp In Pf
1+ NS'L (52N) + 1+
(3.2-16)
where (SI') i equals (S-0/N) (BSi ) • The detection probability
after 3 of 3 post detection integration is
113
1 M,N) D2 (3.2-14)
Plots of Pd versus (90/N) for the baseline configuration
are shown in Section 3.3 for the five target models.
114
3.3 Detection Performance for the Baseline Configuration
The detection performance for the baseline configuration
is presented in this section by developing probability of
detection versus (S0/N) plots. To accomplish this, a cell
false alarm probability before 3 of 3 post detection inte-
gration corresponding to the given system false alarm
specification is first required. Procedures for obtaining
this are presented in Section 2.5. Next, the detection
threshold is determined per Eqn. (3.2-2). For each slant in
a beam dwell, the beam shape loss given in Section 2.4 is
then subtracted in dB from the stated (T0/N) before the con-
ditional detection probability, Pdi' is computed for each
target model. The number of pulses available for integration
for the baseline and modified configurations were determined
in Section 2.3. Finally, the detection probability after
post detection binary integration is determined according to
the procedures outlined in Section 3.2.
Figure 3.3-1 shows the probabilities of detection aver-
aged over the respective target fluctuations assumed for the
five target models described in Section 2.6. For these de-
tection probabilities, the threshold is set such that the
resulting system false alarm rate is 2 per each antenna scan
which takes ten seconds. When (S0/N) is 12.2 dB, the
115
C*4
mmty0
116~
baseline yields the single scan detection probability of
0.32 for a Swerling case 1 reference target when it is lo-
cated at the reference range. Note that this is equivalent
to the cumulative detection probability of 0.9 in 6 antenna
scans as explained in Section 1.1.3. The cumulative detec-
tion probability curves for the five target models are shown
in Figure 3.3-2. Given single scan detection probabilities
of Figure 3.3-1, the cumulative detection probabilities of
Figure 3.3-2 for 6 antenna scans arise as a natural conse-
quence. Therefore, the curves of Figure 3.3-2 are those to
which performance under scan-to-scan processing using 6
scans should be compared. It is evident from these figures
that fluctuating target models yield poorer detection per-
formance than a nonfluctuating target model in the region of
high SNR. Also evident is the influence of correlation
properties on detection performance. Swerling case 1 and
Swerling case 3 are more alike in their detection perfor-
mance than those for Swerling case 1 and Swerling case 2
even though Swerling case 1 and 2 have the same probability
density function for their RCS fluctuations.
Figures 3.3-3 through 3.3-7 are detection performance
plots for Swerling cases 1, 2, 3, 4, and the Marcum model,
respectively, for various cell false alarm probabilities.
With reference to Eqn. (2.5-25), note that an increase by a
117
Cumulative Detection Probabilityof i te anmk
SL 12162
12 16rn ca1e 24
®Swerling case 3Swerling case 2
MHarcum
-4cell false alarm Probability: 7.6xl0
normal time overhead: 5 milliseconds
individual beam shape losses: 0.3/3.43 dB
reference SWR: 12.2 dB
Figure 3.3-2 Cumulative Detection Probabilities for Swerlingand Marcum Target Models - Baseline
118
U),
0u0Cc
0 0 0 0
119c
0 04
N C
oo 0
ko
caPE
L-C"0 i
C4C
120
V
(I)0
02
V0 -a-,-
-U
a
U
Sz a'v~cp~
* zo i *~ :
0 0oS a
"40 Ca,
U(n '4-.
- %~-
0
0cv
'4-,
0)U
a' ~ r- ~O ~ N - 06 o 6 o 6 6 66 6
)wJi~o~ac ~o ~(~~q0qO4~
121
C
I(I)
-W0 4.
CP
0
C4C
122
*m
Q)
-C
.2 -J
*N E
So Cp
cov
1*o In N 0 0 d ~ Cd w
(Pd) U01400100 10 /!!~Oc
123
factor of 10 in the cell false alarm probability results in
an increase of 1000 in the system false alarm rate because
Pa = . Consequently, one must be careful in lowering the
threshold in an attempt to enhance detection. The baseline
performance presented here will provide the basis for mea-
suring performance improvements in the modified configura-
tions discussed in the next two Chapters.
It should be noted that a separate beam shape loss is
used for each slant in the computation of (S/N)i. It is a
common practice in radar engineering to evaluate Pd by using
for each slant the average value of (S/N)i obtained by com-
puting an average value of beam shape loss. However, this
yields optimistic results for Pd of 0.7 dB for Swerling case
1 and 0.4 dB for Swerling case 2 in the vicinity of (S0/N) =
12 dB. These numbers were obtained by comparing two sets of
detection performance results where individual beam shape
losses were used in one set and one average beam shape loss
was used in the other. The results for Swerling case 1, 2,
and Marcum target models are plotted in Figures 3.3-8 and
3.3-9 for individual and average beam shape losses, respec-
tively. This is very important in comparing the performance
of the modified configurations to the baseline performance.
The performance of the modified configurations would appear
pessimistic by the respective amount if one average beam
124
CNNU
.94P4
C14 0
o)
E4
0 to0~
mI ri Z
-4 M
SP VzT, US Q3U0Q;x >~
z 0- %Cl .-
C13 m
0w
o 0 M 0 0 0 0t 0 0 0 0
~6 66 66 66 66 6
not.13a jo SlqlqqoixJ
125
(NN
C~U'
0o 0
C.)
W PW ~01
r 4 Idc~-r041 >
4
D02 0
00
C12N
Z, z
.0I-4
$4
94
00 0) N- CO LO .4 n ' N -
EuoTn3ao jo SI Iioi
126
shape loss had been used in the evaluation of the baseline
performance.
It should also be noted, in Figures 3.3-1 through 3.3-9,
that the normal time overhead per each slant of 5 millisec-
onds was used in the performance evaluation. When a
modified configuration includes a reduction of time over-
head, unless this reduction is a direct consequence of a
certain unique feature of the modification, the measure of
improvement should be by comparison of its performance to
the baseline performance which also includes the reduction
of time overhead. This is so that the improvement imparted
by the specific processing modification can be quantified
separately. For this purpose, plots of the baseline perfor-
mance with a reduced time overhead to 3.5 milliseconds per
each slant are shown in Figure 3.3-10 for Swerling case 1,
2, and Marcum target models.
Additional details on how the two different approaches to
beam shape loss affect detection probability estimates are
included in Section 3.4. What happens to the beam shape
loss when the 3 of 3 post detection integration is changed
to a-2 of 3 post detection integration all within a 3-slant
configuration is also discussed in Section 3.4.
127
04
L)~
~ 00
~w
E-4 m~
E-l 0
4j
zz44
.0
-d I
4pS'T UN DU,. 0
coo
0 C 0~ CO ~ 0 V) 0~ 00 015.
Iolaj o lqeqojzc
128
3.4 Effect of Beam Shape Loss on Detection Probability
In the analysis of detection probabilities for the
baseline 3-slant configuration, it was found that a notice-
ably different performance prediction results when a single
average beam shape loss is used instead of the individual
beam shape loss for each slant. When one average beam shape
loss is used, the resulting detection probability for (90/N)
in the neighborhood of 12.0 dB is more optimistic by 0.4 dB
for Swerling case 2 and by 0.7 dB for Swerling case 1. This
is with the threshold for cell false alarm probability of
7.6x10-4 which, in conjunction with the 3 of 3 post detec-
tion integration, yields the system false alarm rate of 2
for each antenna scan covering a 360 degree azimuth sector
in 10 seconds.
The above result is illustrated for a Swerling case 2
target below. A comparison for Swerling case 1 developed
using numerical integration is included in Section 3.3. The
actual beam shape losses of 0.3 dB for the center slant and
3.43 dB for the two outer slants can be converted to one av-
erage loss by first converting these dB's to equivalent
power levels referenced to a unit power level, obtaining the
average of these power levels, and taking the ratio with re-
spect to unity. The average beam shape loss so computed for
129
the 3-slant configuration is 2.12 dB.
The actual SNR in each CPI, given (S0/N)=12.0 dB is 12.0
dB minus the respective beam shape loss. These are 11.7 dB
for the center slant and 8.57 dB for each of the two outer
slants. When the average loss is used instead, it is 9.88
dB. The corresponding detection probability in the ith
slant for a Swerling case 2 target can be obtained from Eqn.
(3.2-13).
Let the detection probabilities in the center slant and
the two outer slants by Fdl' and 9d2' respectively. Also,
let the detection probability in any slant when the average
beam shape loss is used be denoted by Pd" Letting (S/N)i =
10 §-(dB)/1 0 and using 7.6xi0 -4 for pf in Eqn. (3.2-13)
yields the following detection probabilities:
d - 0.635
Pd2 = 0.416
-Pd = 0.518.
With these probability values, Pd' the overall detection
probability after post detection integration is computed for
the two cases: one using the actual beam shape loss for
each individual slant, and the other using one average beam
130
shape loss for all 3 slants for 2 of 3, and 3 of 3 post de-
tection integration.
2 of 3 Post Detection Integration:
Let Pd(2/3) and Pd(2/3) denote the detection probability
after post detection integration with actual beam shape
losses and with an average beam shape loss, respectively.
Then, Pd(2/3) and Pd(2/3) are given by
Pd(2/3) - 2Pdi Pd2 (-Pd2) + Pd2 2(l-p ) + -d? d2 2 (3.4-1a)
and
P;(2/3) _ 3p2(l -pd) + D3 . (3.4-1b)
dd d d
Substituting values of Pdl' Td2' and Pd previously deter-
mined into the above equations yields
Pd(2 / 3 ) = 0.482
Pd(2/3) = 0.527
Using the average beam shape loss resulted in a detection
probability which is more optimistic by 0.4 dB than the one
with actual losses as explained below. This was determined
by substituting for the (S/N)i in Eqn. (3.2-13) a value 0.4
131
dB less than the one based on the average beam shape loss
and obtaining a Pd value of 0.484. Next, substituting this
value for Pd in Eqn. (3.4-1b) yields a value that is the
same as the one obtained by Eqn. (3.4-1a) using actual
losses.
3 of 3 Post Detection Integration:
Let Pd(3/3) and P*(3/3) denote the detection probability
after 3 of 3 post detection integration with actual beam
shape losses and with an average beam shape loss, respec-
tively. These probabilities are given by
PP/ d3) dld 2 - 0.11, (3.4-2a)
and
P;(2/3)= -D - 0.139 (3.4-2b)d
where use was made of the values for Pdl' Pd2' 1d previously
determined. Again, the result based on the average beam
shape loss is optimistic by an amount of 0.4 dB. This was
determined by following the same procedure used for the 2 of
3 post detection integration example.
Thus, a simplification in computing by use of an average
132
beam shape loss resulted in an optimistic prediction by 0.4
dB for Swerling case 2 targets. An optimistic prediction by
an amount of 0.7 dB resulted for Swerling case 1 targets
when the detection evaluation is based on 3 of 3 post detec-
tion integration. This difference is obtained by comparing
the numerical integration results of Eqn. (3.2-11) for
Swerling case 1 with two different approaches to the beam
shape loss. The difference in detection prediction was also
computed for the Marcumtarget model. These results were
plotted in Figures 3.3-8 and 3.3-9 in Section 3.3 using in-
dividual and average beam shape losses, respectively.
Next, the following questions is considered: What happens
to the beam shape loss when the post detection integration
rule is changed from 3 of 3 to 2 of 3 in a 3-slant con-
figuration? There is a common misconception that the beam
shape loss would be reduced under a 2 of 3 rule. This notion
is based on the observation that if the binary post detec-
tion rule requires only two hits, the two contiguous slants
that could provide these hits would be found more favorably
centered about the peak of the antenna beam than would be
for 3 slants. While this observation is correct, the detec-
tion rule under this situation approaches a 2 of 2, not a 2
of 3 rule since the 3rd (and the 4th) slant is too far from
beam center to contribute significantly. The detection
133
probability based on a 2 of 2 rule with less beam shape loss
is inferior to a 2 of 3 rule with slightly higher beam shape
loss. Alternatively, one can correctly observe that there
are two 3-slant sets on a sliding window providing 2 of 3
detection opportunities when 4 slants are symmetrically
situated about the peak of the beam even though the beam
shape losses for the two outer slants are high. This obser-
vation is again correct. But, the 4 slants with higher beam
shape loss produces inferior performance, as will be shown,
to that produced by 3 slants symmetrically situated about
the peak of the beam.
Fortunately, the differences can be readily quantified.
To this end, a snap shot of the train of modulation periods
passing by a point target at an instant of the most
favorable placement for 3 slants, and 2 (or 4) slants are
depicted in Figure 3.4-1 (a) and (b), respectively.
Q of beam Q of beamtoward a oint target toward a point target
LBSi1) 3.40 0.30 3.40 7.29 1.08 1.08 7.29
Pdi Pd2 Pdl Pd2 Pd2 Pd2 Pdl Pd2
(a) 3 slants (b) 2 or 4 slants
Figure 3.4-1 Beam Shape Loss for 2 of 2, 2 of 3 and 2 of 4Processing given 3 Slants in a Beamwidth
134
The beam shape loss for each slant is computed according to
the method described in Section 2.4 and is indicated under
the slant for which it applies in the Figure.
As before, the detection probability in each slant for
the 3-slant case is denoted Pdl for the center and Pd2 for
the two outer slants. For the 4-slant case, they are denoted
Pdl for the two inner slants and Pd2 for the two outer
slants. The Swerling case 2 model is again used for simplic-
ity to determine the detection probabilities based on the
calculated beam shape losses so as to establish which place-
ment of the slants results in the minimum loss. We are
still dealing with a 3 slant configuration meaning there are
only 3 slants in a half power beam width and the detection
rule is 2 of 3 hits from a set of 3 contiguous slants se-
lected on a sliding window which advances at one slant in-
crement. An overall detection probability given 4 slants
and using a 3-slants at a time sliding window with a 2 of 3
rule is first established. Since the two 3-slant sliding
windows are overlapping, the probabilities of detection are
not independent, and a careful sorting is required before
determining the overall detection probability. This is best
accomplished by establishing the matrix of all possible com-
binations of only 2 of 4, only 3 of 4, and 4 of 4 detection
possibilities within the 4 slants given, and eliminating the
135
ones which do not have the 2 or 3 required hits within the 3
slant sliding window. Note that an M of N means M or more
hits in N trials while only M of N means exactly M hits in N
trials. This matrix of hits and misses along with the slid-
ing window is shown in Figure 3.4-2.
Now, the expression for the probability of detecting 2
of 3 in 4 slants on a 3-slant sliding window can be written
down by inspection of the matrix of Figure 3.4-2. The num-
ber of ways a detection can occur is all the possible combi-
nations shown in the Figure less those excluded by the
3-slant sliding window test. The sum of different numbers
of ways a detection can occur multiplied by the correspond-
ing probabilities gives the overall detection probability:
Pd(2 / 4 ) (on a 3-slant sliding window)+-d - p 2 ) + f 21
diPd2 ( 1 dl)(|Pd2 I d2 d 2 d2 d2 1 d 2
Obviously, the probability of detecting 2 of 2 from a
2-slant set is
- 2Pd(2/2) =, i 2
The probability of detecting 2 of 3 in a 3-slant set was de-
rived before in Eqns. (3.4-la,b).
136
center lineof beam
Iouter slants case rejected byinner slants 3-slant window
I I i I1 way to get
4 hits in 4 slants x x x x
x x x 04 ways to get x x o x x
3 hits and 1 miss x o x x xin 4 slants 0 x x x
x x 0 0x 0 x 0
6 ways to get x 0 0 x x2 hits and 2 misses o x x o
in 4 slants o x 0 x0 0 X X
prob. of detection Pd2 Pdl Pdl Pd2
I I
3-slant sliding windows
Figure 3.4-2 Matrix of Possible Combinations of 2 of 4Detections on a 3-Slant Sliding Window
137
The probabilities of detection after 2 of 2, 2 of 3, and
2 of 4 binary post detection integration are computed as a
function of (S0/N) according to the equations derived for
the Swerling case 2 model. The results are listed in Table
3.4-1. It can be seen from the table that the detection
probability based on the 2 of 3 rule with 3 slants centered
about the beam center is superior to the result based on 2
of 2 or 2 of 4 rules using 2 slants or 4 slants centered
about the peak of the beam. This is in spite of the smaller
beam shape loss for the center 2 slants in these cases. As a
final point, it is noted that the average beam shape loss
depends on the number of slants within a half power
beamwidth. It has nothing to do with the values of M and N
selected for M of N binary integration. Those slants fall-
ing outside of the half power beamwidth contribute very
little to detection performance in the small signal region.
Actually, Blake [7] determined that the optimum beam utili-
zation is 84 % of the half power beamwidth.
138
Table 3.4-1
Detection Probabilities with 2 of 2, 2 of 3, and2 of 4 Processing for a Swerling Case 2 Target
10.4 .222717 .308167 .312528
10.5 .236682 .328043 .332856
10.8 .250970 .348318 .353562
11.0 .265551 .368923 .374578
11.2 .280395 .389785 .395834
11.4 .295472 .410828 .417259
11.6 .310748 .431975 .438782
11.8 .326191 .453147 .460333
12.0 .341769 .474263 .481840
12.2 .357450 .495247 .503239
12.4 .373202 .516021 524463
12.6 .388994 .536512 .545452
12.8 .404797 .556649 .566148
13.0 .420580 .576367 .586496
13.2 .436315 .595605 .606448
13.4 .451976 .614308 .625959
13.6 .467538 .632429 .644990
13.8 .482975 .649925 .663504
* 2 of 4 on a 3-slant sliding window as explained
in the text.
139
CHAPTEL 4
SYSTEM PERFORMANCE WITH SCAN-TO-SCAN PROCESSING
A description of scan-to-scan processing (SSP) and a de-
tailed analysis of the modified radar configuration with
scan-to-scan processing is presented in this Chapter. A
considerable amount of effort has been spent in industry in
seeking a detection performance improvement by means of
scan-to-scan processing. Surprisingly, however, there is
little written material on scan-to-scan processing in the
open literaturelin spite of more than a decade of industrial
independent research and development efforts that went into
investigation of the concept. Evidently, the only documenta-
tion that exists are proprietary internal company
reports not available to the general public. These reports
document interim results of trials with various flight test
samples and some simulations. Optimistic improvement claims
as much as 12 dB in equivalent signal-to-noise ratio are
made without accompanying analytical verification Similar
work named Track before Detect (TBD) [11, 12] was carried
out under the auspices of RADC.3 This is not to be confused
with the TBD term used in the moving target indication
algorithm in infrared imaging [13].
1. as applies to airborne surveillance radars2. Upon careful examination, this 12 dB includes improvement projections
arising mostly from other radar processing changes not directlyattributable to SSP.
3. for a ground surveillance radar
140
In view of the primarily experimental nature of
industry's investigation, a need for' an independent
theoretical review of the concept arose as a consequence.
This investigation provides one such review. The analysis
presented will show that only a marginal improvement is di-
rectly attributable to SSP when correct comparisons are
made.
4.1 Description of Scan-to-Scan Processing
As the effective range of avionics systems continually
increases, so does the need for increasing the effective
range of sensor systems. Because of the enormous cost and
risk involved in developing and deploying a new radar
system, increasing emphasis is being placed on improving a
proven existing system capitalizing on recent technology
breakthroughs in signal processing. Thus, scan-to-scan pro-
cessing became one of the most appealing concepts for
improving performance. The improvement sought is to maintain
the same detection capability for targets whose radar cross
section (RCS) is continually being reduced. Equivalently,
an alternate aim is to extend the detection range for
conventional targets.
141
As described in Chapter 3, detection performance is di-
rectly related to the threshold setting. The philosophy
behind SSP is to lower the threshold to enhance detection.
The resulting increase in system false alarm is expected to
be managed through data processing over many radar scans
whereby detection histories of true and false targets in
past scans are correlated and those detections which do not
result in realistic trajectories are rejected as false
targets. According to this notion, there is no limit as to
how much the achievable improvement can be. It is only a
matter of how much data processing is to be performed to be
able to reject all false targets.
On the other hand, there is a theoretical approach for
determining the upper bound on performance improvement which
is possible through SSP. Recall Barton's result on
surveillance radar performance quoted in Section 1.1.3 when
the surveillance objective is to detect a target within a
specified interval of time with a high probability of
success. Given the total number of pulses available for
integration during this time interval and an option to
adjust the scan rate, Barton compared the resulting
performance between using the available pulses all in a
single scan with a slow scan rate and, alternatively,
dividing these into many scans with a faster scan rate. He
142
considered the integration loss, scan distribution loss, and
target fluctuation loss, and concluded that it was more ad-
vantageous to use more than one scan to achieve the goal.
However, his analysis is based on some ideal assumptions,
e.g., a stationary target which falls in the same radar
resolution cell in every scan, and operation of the radar in
a low PRF mode sudhAthat division of the available number of
pulses into many scans does not increase the total time
overhead.
Barton's integration loss is the loss incurred in video
integration when compared to what can be achieved in an
ideal coherent integration. His scan distribution loss is
the additional number of pulses that are required for a
nonfluctuating target using many scans to achieve the same
detection probability as for a nonfluctuating target using a
single scan. The fluctuation loss is the additional
signal-to-noise ratio required, on the average, for a fluc-
tuating target in a single scan to achieve the same level of
detection probability as for a nonfluctuating target. In
Barton's multi-scan strategy, detection probability in each
scan is assumed independent from any other scan.
Suppose now a storage device is introduced to accumulate
the results of several successive scans and a hit is
143
declared when the same target is detected in the same
resolution cell in J out of K successive scans. This, then,
would reduce the scan distribution loss. This, in essence,
is the scan-to-scan processing concept. In terms of math-
ematical expressions, given a total number of K scans in the
baseline without SSP, the cumulative detection probability
Pc, that is the probability of detecting the same target in
one or more scans, was given as
PC M _-(l-pd) K (4. 1-1)
where Pd is the single scan detection probability assumed to
be independent from scan to scan. With SSP, the cumulative
detection probability, that is the probability of detecting
the same target in J out of a total number of K scans, is
given by
K K! Pi(l-Pd)K (4.1-2)P J fi i i(K-i)l
Of course, both detection rules should be subject to the
same system false alarm requirement. For J>l, the effect of
SSP is to lower false alarms relative to the J=l case.
Therefore, for maintaining the same levels of system
false alarms, the detection threshold can be lowered with
SSP thus enhancing the detection probability.
144
The extent to which the threshold can be lowered for a
moving target, with the resulting large increase in interim
system false alarms per scan prior to SSP, depends on the
size of the correlation window which determines the number
of cells per scan where reports are correlated with those of
other scans. If sufficient information about the target
trajectory is known such that correlation could be performed
on a single resolution cell for each scan over the number of
scans in which correlation is performed, false alarm sup-
pression by the J of K integrator in SSP would be great.
However, the likelihood is small of correctly estimating the
resolution cell for each scan in which the target is lo-
cated. To account for the radar platform motion and target
maneuver during the scan-to-scan correlation period, the
correlation window for each scan is made larger for succes-
sively earlier scans as shown in Figure 4.1-1.
Note that, even with SSP, M of N post detection integra-
tion still takes place within each scan so as to resolve
target range ambiguity at least partially in each scan. Oth-
erwise, target range (and velocity) information would not be
available and there would be no basis to correlate present
scan detections with those of the past scans.
145
correlation window forthe Kth scan
correlation window forthe ith scan
resolution cell in which targetis reported in the current scan
Note: Azimuth and range correlation windows only are shown.Velocity correlation window is not shown.
Figure 4.1-1 Correlation Windows for Successively EarlierScans
146
Two specific SSP concepts are examined in this investiga-
tion. One uses the 3-slant configuration, as in the
baseline, and employs a modified version of scan-to-scan
processing. As a result, this approach is referred to as
the modified J of K SSP. The other is based on a 2-slant
configuration and uses conventional J of K SSP. In both
versions, a sliding window is used with respect to the K
scans such that a decision is made after each scan.
In the modified J of K SSP, the threshold level is kept
the same as in the 3 of 3 baseline configuration and a full
success is declared when threshold crossings occur in all 3
slants of a given scan. All range and velocity ambiguities
can be resolved and operation is the same as for the
baseline configuration. Hence, SSP is bypassed for a scan in
which a full success occurs for the given target in that
particular resolution cell. A partial success is declared
within a single scan if and only if 2 threshold crossings in
any of the 3 slants in a particular doppler filter occur.
Note that this constitutes a failure in the baseline 3 of 3
post detection rule. In the modified J of K SSP rule, a
detection is declared when either a full success occurs in
the present scan, which is the same as the baseline, or both
a partial success occurs in the present scan and (J-l) or
more partial or full successes have occurred in the past
147
(K-i) scans. For J=3 and K=8, this SSP is referred to as
the modified 3 of 8 SSP. For the modified J of K SSP, the
probability of detection at the end of each scan is called a
hybrid single scan detection probability since it can arise
from either a full success which is independent from
scan-to-scan or from partial successes in a number of scans
which makes the Pd based on them no longer independent from
scan-to-scan. Since the threshold level is kept the same as
in the 3 of 3 baseline configuration, the 2 of 3 post detec-
tion integration generates in each scan an increase in false
alarms of more than 3 orders of magnitude. The association
from scan-to-scan is an attempt to reduce the false alarms
to an acceptable level.
The conventional J of K SSP concept is based on a 2-slant
configuration. By using 2 slants within a scan, it is pos-
sible to integrate more pulses per slant although the number
of slants is reduced. In this concept, the threshold level
is adjusted to maintain the same false alarm rate per scan,
before SSP, as in the modified J of K SSP described above.
This is done so that the two concepts can be compared on a
common basis. When there are only 2 slants, 1 of 2
detections cannot be considered as a partial success because
range and velocity information is unavailable for
correlation purposes due to unresolved ambiguities. Also,
148
2 of 2 detections cannot be treated as a full success be-
cause the false alarm rate would be too high and the range
ambiguity could only be partially resolved. For conventional
J of K SSP, J or more partial successes in K scans are
required to declare a detection. Because of the sliding
window, a decision is made after each scan. However, the
detection probability associated with these decisions should
be compared with a cumulative detection probability based on
K scans as opposed to a single scan detection probability.
4.2 The Scan-to-Scan Correlation Window
The size and position of the correlation window allocated
for each scan is a function of the target parameters. In
particular, the correlation window for each scan is designed
to be large enough such that a maneuvering target appears
within the window. For this purpose, it is necessary to
account for the target's tangential velocity and
acceleration. In determining the window size, it is assumed
that the maximum linear acceleration is 1-g (i.e., 9.8
m/sec.) along the radial velocity vector, and the maximum
tangential velocity is given by
149
VT . {5VR 5VR<Vmx
Vmax 5V>Vmax
where VR is the measured radial velocity in the present scan
and Vmax is the maximum possible velocity assumed.
The geometry for the correlation window in the ith scan
is shown in Figure 4.2-1. The procedure consists of
selecting a candidata target report which has the potential
of being included in the window, constructing the window
ussing its position and that of the present target report
under test, rejecting the candidate target report if it
falls outside of the window, and retaining the candidate
target report if it falls inside. This procedure is repeated
for all candidate target reports in the ith scan. With ref-
erence to Figure 4.2-1, define the following notation:
WR: range correlation window width along the radial direction
WT: cross range correlation window measured normal to theradial direction
VR: measured target radial velocity
VA: platform velocity
al: linear acceleration of the target along its radialvelocity vector
0i. measured target azimuth angle in ith past scan
150
candidate target report
target trajecto
present target report under testwith measured range and velocityin the present scan
platform latform position in the ith pastflight path -san -
present platform position with velocity
VA
Figure 4.2-1
Correlation Window for the ith Scan
151
e0 : measured target azimuth angle in the present scan
R, measured range to the target in the present scan
RA: - radial distance from the platform position to the pointwhere the radial lines to target positions in the present
scan and ith past scan intersect
i AR: estimated difference in range to the target between the
present scan and ith past scan (AR=iTFVR)
Also, introduce the notation:
ar: range measurement error
a.,: azimuth angle measurement error
ap : platform position error
For the ith scan,
iAR = iTFV R
where TF is the frame time. From the law of sines, the dis-
tance RA is related to target azimuth angles by
RA VA(iTF)
sin(ir-Oi) - sin(Oi_90) ; i>0
RA VA(iTF)sini f sin(Oi-e0 ) ; i
For the ith scan, the spatial size of the correlation window
extends over
152
± WR =[(ai(iTF)'/2)2 + 2(oR)2 + 2(') 2 + 2jR+RA+i AR)oesin4el-0) 2] 1/2
± WT h[(VT(iTF))2 + 2[(R+RA+iAR)a, 2 + 2(ap) 2 ]1/2
Because of variations and uncertainties in the present
target velocity, it is necessary to extend the correlation
window into the velocity domain. The correlation window size
for the velocity is
± WV =[(a(iTF))2 + (V jrsi e-e)2 ] 1/2
Having determined the (K-l) correlation windows related to
the present target report, the next step is to calculate the
false alarm multiplier associated with these windows. Let ni
denote the number of resolution units in the correlation
window for the ith back scan. The false alarm multiplier
for the (K-l) correlation windows is given by
Mk = (nl)(n 2)- - - (nK-1)
With SSP the total number of false alarm opportunities
equals MK times the conventional number of false alarm op-
portunities for a radar without SSP.
To determine a realistic value for MK, it is necessary to
specify a typical target maneuver For this purpose, con-
1. as worst case
153
sider a target making a 6-g coordinated turn with a speed of
450 knots at a range of 150 nautical miles. The maneuver
scenario is illustrated in Figure 4.2-2. Also, let
VA - 360 knots
a = 0.8 nmi
a- 0.003 radians
01- 0.04 nmi
For this maneuver, the flight path is circular and a1 = 0.
The corresponding correlation window size, in terms of
azimuth angle expressed in degrees and the number of range
and doppler cells, is listed in Table 4.2-1. On the
average, there are 3 slants contained in 1 degree azimuth.
The number of resolution units, ni, for the ith scan is
listed in the column labeled ni(l) where a resolution unit
is 1 slant by 1 range cell by 1 doppler cell. The false
alarm multiplier for K scans, K = 2,3,--,8, is listed in the
column labeled M(I).
If it were possible to estimate past target positions
down to a single resolution cell for each back scan, then,
the false alarm multiplier arising from SSP would be unity.
As shown in Table 4.2-1, the false alarm multiplier in-
creases dramatically as the window size is made large enough
so as to yield a high likelihood of covering the target.
154
2Turn Radius: r V /a
Centripetal Acceleration: a
Turn Rate: y - Vt/r
Target Position: X- X0 - r cos yManeuver Y - Yo + r sin TCenter
/(Xo.Yo) Target Range: R * ((X - Vat) 2 + y2 h
Range Rate: R(t) A R(t) - R(t At)At
ManeuverRange
a- -- '-X (Longitude).. platform moving @ 360 kts
Figure 4.2-2 Maneuver Scenario used for testing the SSPCorrelation Window
155
-H-V v- LN o% 40i Coi r c,
C14 0. 0% (-. O ~ j 0% 0(A~0 %.& 0I - I 4 % 0
0- -P4 m m * n r:
V4 04
0. w- .. 0 0
Ln C-~. -4 v-I v-I v v v-I
~IH 044 OD V n t
.4 ~ ~ . v~ U7 N N In vi
o- 0~ r-4 0 I 0 r- 4
0 0 tr eI N mn 04 r- CI 0-H r. I m* vt- %o r Nm
oq %o r . o . -
0 0W
6w4 .- Ic 4 m 0 v n %D r. c
-H I' rv*f Nw I n 0 r,!4 0 ~ 0 4 vtoNo
(a - *.
* Q q ~ I v44 00156 0
Because this number is unacceptably large, a bigger resolu-
tion unit composed of 3 azimuth slants by 10 range cells by 6
doppler cells will be used in this analysis. (Any grouping
which is reasonable for the particular application can be
used.) Multiple threshold crossings in this larger resolution
unit now count as one. The corresponding number of resolution
units is listed under the column labeled ni(2) and the result-
ing multiplier is listed under the column labeled MK(2). Ob-
26 11serve that M8(1) = 8.28x02 while M 8(2) = 1.35x1011 Even
though M8(2) is 15 orders of magnitude smaller than M8(1), it
is still a very large numbnr.
157
4.3 Determination of threshold Setting and Depth of Scan
from False Alarm Considerations
The procedure for determining the cell false alarm prob-
ability pf from the system false alarm specification was de-
veloped for a general M of N post detection integrator in
Section 2.5. This procedure requires determination of the
false alarm time, which follows directly from the system
false alarm specification, and the determination of false
alarm opportunities which takes into account utilization
factors for time, range cells, and doppler cells, and cor-
relation factors due to range and velocity unfolding for am-
biguity resolution, and use of sliding windows for M of N
binary integration. From the false alarm time Tfa and the
number of false alarm opportunities in a false alarm time n,
the false alarm probability after post detection integration
Pfa is determined. From the knowledge of the specific binary
post detection rule, the cell false alarm probability before
the post detection integration pf is then determined from
P fa '
In the modified J of K SSP in a 3-slant configuration,
SSP is invoked if and only if a threshold crossing occurs in
2 of the 3 slants. This is called a partial success. When
threshold crossings occur in all 3 slants, which constitutes
158
a full success, SSP is bypassed. The modified J of K pro-
cessor then becomes identical to the baseline processor.
For this bypass feature to perform as in the baseline, the
threshold setting for each slant should remain the same as
that set for the 3 of 3 processing in the baseline. This is
equivalent to a lowering of the threshold for the 2 of 3
processing such that the false alarm rate per scan before
SSP increases by 3 orders of magnitude, as shown in Table
2.5-1.
To demonstrate the 3 orders of magnitude increase in the
false alarm rate before SSP and to facilitate evaluation of
the false alarm rate after SSP, the following events
relative to false alarm occurrences are defined:
Fl: false alarms occur in all 3 slants of a scan
F2: false alarms occur in 2 of the 3 slants of a scanwith no false alarm in the remaining slant
F3: false alarm occurs in at most one of the 3 slants of
a scan with no false alarm in the remaining slants
F4: event F1 for the present scan
F5: event F2 for the present scan
F6: event Fl or F2 for J-1 or more of the past K-1scans and event F3 in the remaining scans
F7: intersection of event F5 and F6.
159
Let n(Fl) and n(F2) denote the number of false alarm op-
portunities in a false alarm time associated with F1 and F2,
respectively. The average number of false alarms per scan
before SSP is then given by
FA/scanibefor, sSp P(F1)n(F )Z. + P(F2)n(F2) T- (4.3-1)f a'l- f S2
The first term in Eqn. (4.3-1) is the average number of
false alarms in a scan for the baseline which is specified
to be 2. The second term results by allowing partial suc-
cesses to also be counted. Note that
P(F2) -T-I! p (l-pf) -3f 2. (4.3-2)
In Section 2.5 it is shown that the cell false alarm rate,
pf, equals 7.6xi0-4 for the baseline. Consequently,
P(F2) = 3p2 =1.73x0 "6 . (4.3-2a)
f
Analogous to the derivation of Eqn. (2.5-32), the total num-
ber of false alarm opportunities per scan for event F2 is
F T F nr(frWf2)nfPrnmsf 2
rW ReT TF
(nr+rB) -fa
160
For the system under consideration
N-42, Wf2=2 .2, nf=195, RT-6173, rB- 3 , n T=n F- =1040.
f 2
It follows that
F -= 2.40x10 . (4.3-2b)
Hence,
TF
P(F2)n(F2)T- T (1.73x10")(2.40x109) -4.15x10 . (4.3-2c)
The large predicted number of false alarms before SSP agrees
with the flight test result illustrated in Figure 4.3-1.
Having introduced the false alarm multiplier MK(2 ) in
Section 4.2 to account for the expanding correlation window
as a function of scan depth, the analysis can be carried out
by 1) considering correlation between single resolution
units over K scans for which the false alarm multiplier is
unity, and 2) applying M(2) to account for the actual cor-
relation window. With respect to step 1, the size of the
correlation window in each scan is one resolution unit. The
corresponding SSP process can be thought of as a combined
experiment of K identical and independent subexperiments
where the kth subexperiment, k = 1, 2, - - , K, consists of
161
Curr ent display is ~jEtde,Scans I through 57 tumber oF reports 47224522 to 22:5!50
Lats 22.1 to 29 8 to3 -882 tv -UO./6
.
!A
I F1n 3nt3 400
C r .1nd s abIize d r atqe
21,134 false alarms / 57 scans= 371 false alarms / scan over a 34 degree azimuth
sector
(371 x 360/34)= 3,928 false alarms per scan before SSP
predicted number of false alarms / scan before SSP= 4,150
Figure 4.3-1
Flight Test Verification of the Prediction of False Alarmsbefore SSP with 2 of 3 Post Detection Integration
162
m =lmm ram om=numnln nmwn ~ nmnnunnunnnnmunuan ulnmu a
a detection trial in the kth scan for the resolution unit in
question. The sample space S for the combined experiment
can be represented by the product space
S = S1 X S2 X S3 X -- X SK (4.3-3)
where the kth sample space, Ski can be partitioned into the
three mutually exclusive events, Fl, F2, and F3. Let the
outcome of the kth experiment be denoted by the event
Ak where Ak is either Fl, F2, or F3. Then, the outputs of
the X subexperiments can be represented in the product space
as
A1 x A 2 x A 3 x -- x AK
For the case of statistical independence,
P(A1 x A 2 x A 3 x -- x AK) = P(A1)P(A2) -- P(AK) (4.3-4)
That is, with independent experiments, the probabilities for
events defined on S are completaly determined from prob-
abilities of events defined in the subexperiments [5].
By the rule established, a 'hit' in the modified J of K
SSP is a union of events F4 and F7. The probabilities of
these events are
163
P(F4) = P(F 1) (4.3-5a)
P(F7) = P(F5 n F6) -P(F2)P(F6) ,(4.3-5b)
respectively. Note that
P(F 1) = p3 (4. 3-6a)f
P(F2) -4 P(-f 3P 3-3ps = 3p2 432
K1i (K-i1)! --P(F6) - i!K Ej),P(F 1) +P(F2j [P(F3) K-i (4.3-6b)
The equivalent cell false alarm probability after SSP is
given by
P foafe S = P(F4 UF7) .(4.3-7)
Since events F4 and F7 are mutually exclusive,
P "aferSS 2 P(F4) + P(F7) = P(F4) + P(F5)P(F6)
- P(F1) + P(F2)P(F6) . (4.3-8)~
P aarSP is referred to as an equivalent cell false
alarm probability because it applies to the situation in
which the size of the correlation window in each scan is one
164
resolution unit. The system false alarms after SSP are now
determined by suitably modifying Eqn. (4.3-1). The first
term remains unchanged since this accounts for the SSP by-
pass. The second term is modified by replacing P(F2) with
P(F7) and multiplying the result by the false alarm multi-
plier MK(2). Consequently, the average number of false
alarms in a scan after SSP becomes
FA/scan1ar SSP = P(F1)n(FI)T E + P(F7)MK(2)n(F2TfE (4.3-9)Tf . 1 f &2
The first term in Eqn. (4.3-9) is equal to 2 false alarms
per scan. Recall that it is required to be 2 or less by the
baseline specification. For the modified J of K SSP to meet
the same specification, the second term in Eqn. (4.3-9) must
be negligible. With pf=7.6xl0-4 , it follows from Eqns.
(4.3-5b), (4.3-2), and (4.3-6b) that
I.08x10"16 for J=3, K-8P(F7) =
15.18x10-17 for J=3, K=6
The number of false alarm opportunities, [n(F 2 )TF/Tfa ], was
found in Eqn. (4.3-2b) to be 2.40xi09 . From Table 4.2-1,
the false alarm multiplier MK(2) is
I.35x10 1' for J=3, K=8
I 2.79x106 for J=3, K=6
165
Thus, the second term in Eqn. (4.3-9) yields
(1.08x10")(I.35x10 1 )(2.40xl09 ) = 3.49xO0Ifor J=3, K=8P(F7)MK(2)[n(F2) Tf
f 2
(5.18x10- 17 )(2.79xl0 6)(2.40xId9 ) = 0.345for J=3, K=6 .
The above results indicate that a suitable choice for the
target scenario described in Section 4.2 and the modified J
of K SSP is J=3, and K=6.
The conventional J of K SSP is considered next. For this
SSP, the number of slants in a beam dwell is changed from 3
to 2 in the hope of enhancing detection probability. This
change is intended to increase the SNR per slant by increas-
ing the number of pulses coherently integrated. The overall
time overhead per each beam dwell is also reduced because of
one less slant. The threshold is set, somewhat arbitrarily,
such that the resulting false alarms in a single scan before
SSP are equal to those arising from the 2 of 3 post detec-
tion integration used with the modified J of K SSP, as given
in Eqn. (4.3-2c). This enables comparison of the two SSP's
on a common ground.
The average number of false alarms per scan before SSP in
166
the modified J of K SSP with 2 of 3 post detection integra-
3tion is approximately 4.15x10. This is equivalent to the
false alarm time of
TF = 2.41XI0- $ secondsTfs = FA/Scanbefore SSP
Now, two additional false alarm events are introduced:
F8: false alarms occur in a scan in both slants of a2-slant configuration
F9: event F8 in J or more of the K scans.
A false alarm occurs after SSP only if event F9 occurs.
An expression for Pfa(M/N) is given by Eqn. (2.5-24).
Letting M = N = 2, it follows that
Tf,,B eI( iP-. = P(F8)- T- r (K1f -N j=1
Substituting the numerical values found in Table 2.5-1 gives
P-1= (2.41x10-3(l.25x10 6) (0.479) (0.933) (2) (0.762) (282) (2)fa 2
- 7.89x10 .
-6. 2Hence, P(F8) equals 1.27xi0 - . Since P(F8) = pf, pf equals-3
1.13x10 .
167
Let n(F8) denote the number of false alarm opportunities
in a false alarm time associated with event F8. The average
number of false alarms per scan, which is equal to 4.15x10 ,
can be written as
FA/S~fl~.tr. -TF
FA/scan, before SSP - P(F8)fn(F8)].T- - 4.15x10 3
It follows that the number of false alarm opportunities in a
scan is
[n(F8)TF 3.27xi09 . (4.3-10)
On the other hand, the number of false alarms per scan after
SSP is
FA/scani ft.r SSP - P(F9)MK(2)[n(F) • (4.3-11)
P(F9) is given by
K W1K-P(F9) = K P(F8)'(1-P(F8)) K '
2.41xl-"n for J=2, K=6
4.09x10 - for J=3, K=6
From Table 4.2-1, the false alarm opportunity multiplier for
6a scan depth of 6 is 2.79X106. Substitutions of these num-
bers into Eqn. (4.3-11) yields
168
2.19xlO for J=2, K=6FA/scanj.ft. s = .90.37 for J=3, K 6
It is evident that 2 of 6 SSP does not meet the system
false alarm requirement. Suitable values of J and K for the
target scenario described in Section 4.2 and the conven-
tional J of K SSP are J = 3, K = 6.
169
4.4 Detection Performance under Modified J of K SSP
Detection performance under the modified J of K SSP using
a 3-slant configuration with 2 of 3 post detection integra-
tion is evaluated in this Section. The performance evalua-
tion is based on the cumulative detection probability under
a one minute time constraint. Let L denote the number of
scans within this constraint. For a 6 revolution per minute
antenna rotational rate for 360 degrees azimuth coverage,
the value of L equals 6. Recall from Sections 4.2 and 4.3
that extending the depth of correlation in SSP beyond 6
scans results in the average number of false alarms grossly
exceeding the system specification when adaptive correlation
windows are designed to accommodate target maneuvers up to
6-g coordinated turns at moderate speeds. Thus, even with-
out the time constraints, associating targets beyond 6 scans
would be impractical if not impossible, when ordinary target
maneuvers are considered. In addition, the problem that
arises in a dense target environment is the possibility of
incorrect correlation of target tracks.
As with the computation of system false alarms presented
in Section 4.3, the following events are defined to fa-
cilitate the analysis:
170
Dl: detections occur in all 3 slants of a scan
D2: detections occur in 2 of the 3 slants of a scanwith no detection in the remaining slant
D3: detection occurs in at most one of the 3 slants ofa scan with no detections occurring in the remain-ing slants
D4: event Dl for the present scan
D5: event D2 for the present scan
D6: event Dl or D2 for J-1 of the past K-1 scans
D7: intersection of event D5 and D6
D8: event Dl in one or more of the L scans
D9: event D2 without event Dl in J or more scans of aK scan deep sliding window where the window is oneor more subsets of the L scans
D10: complement to the union of events D8 and D9
DI1: event D2 without event D1 in J or more of theL scans
D12: complement to the union of events D8 and Dll
D13: J or more partial successes occurring in L scanswithout the occurrence of a full success but thesepartial successes do not occur within a K scandeep sliding window.
The probability of detection is a conditional probability
which is conditioned on the hypothesis that a target is
present. Consequently, it is assumed for SSP that the
resolution cell occupied by the target in each scan is known
a priori. Therefore, the correlation window used in false
alarm calculations is not applicable to evaluation of the
detection probability.
171
As was done for false alarms in Section 4.3, cumulative
detection can be viewed as an event in a combined experiment
having a sample space S. The combined experiment without
SSP consists of L independent and identical subexperiments,
each with its sample space SL, where the Ith subexperiment
is a detection trial in the Lth scan for the resolution cell
in question; = , 2, --, L. The combined sample space is
given by
S=S! X S2 X X xSL.
The Lth sample space can be partitioned into the three
mutually exclusive events Dl, D2, and D3. Let the output of
the Jth experiment be denoted by the event Bt where B4 is
either Dl, D2, or D3. Then, the outputs of the L subex-
periments can be represented in the product space as
BI xB2 x -- x BL.
For the case of statistical independence,
P(Bi xB2 x -- x BL) = P(B1)P(B2) -- P(BL) .
That is, with independent experiments, the probabilities for
events defined on S are completely determined from prob-
abilities of events defined in the subexperiments.
172
Without SSP the detection in each scan is independent
from scan to scan. Let Pd denote the detection probability
for a single scan. The cumulative detection probability Pc
is defined to be the probability of detecting a target in
one or more of the L scans. Assuming that the range closure
during this time span is negligible, Pc is given by
PC = I-(1-Pd) L . (4.4-1)
With the modified J of K SSP the cumulative detection
probability cannot be determined from the single scan detec-
tion probability since correlations are involved with detec-
tions in previous scans. As a result, independence no
longer exists from a detection in one scan to the next. Un-
der the modified SSP rule, recall that a 'hit' is declared
in a scan if the event (D4 U D7) occurs. However, because
of the above mentioned dependence between detection trials,
the cumulative detection probability over L scans is not
readily determined in terms of P(D4 U D7). The latter prob-
ability is referred to as the hybrid detection probability
after SSP and is given by
Pd(hyb) = P(D4) + P(D7) . (4.4-2)
A little thought leads to the conclusion that the cumulative
173
detection probability for the modified J of K SSP is the
probability of the event (D8 U D9). Note that events D8 and
D9 are mutually exclusive events. Hence,
D8 n D9 = 0 (4.4-3)
where 0 denotes the null set. Also, because D10 is the
complement of D8 U D9,
D8 U D9 U DIO =S .
In addition, note that Dl, D2, and D3, which form a parti-
tion of the sample space Sj, are also mutually exclusive.
For convenience, P, p, and q will be used to designate the
probabilities P(Dl), P(D2), and P(D3), respectively. It fol-
lows that
P = P(DI) = Pd( 3/ 3 ) (p) (4.4-4a)di d2
p - P(D2) =-2pP(1-P ) + p2 (-P (4.4-4b)p dl d2 d2 d;2 'drs(44b
q = P(D3) = [(I-Pdl)(-Pd2) 2+2pd2(I-Pdl)(I-pd2)+Pdl( -Pd2)1]
= 1 - P -p . (4.4-4C)
where the overbar denotes averaging with respect to target
RCS fluctuations. The detection probability in the ith
slant Pdi and the method of averaging are developed in Chap-
174
ter 3.
From the above discussion, the cumulative detection prob-
ability for the modified J of K SSP is
PCIW/ SSP P(D8 u D9)
= I - P(DlO) (4.4-5)
Since D8 and D9 are mutually exclusive, Eqn. (4.4-5) becomes
Pc Iw/ P = P(D8) + P(D9) . (4.4-6)
Observe that
LL
P(D) - 1 - Pd(3/3)) =1 - (P + qL. (4.4-7)
This corresponds to the cumulative detection probability for
the baseline configuration. Therefore, the second term in
Eqn. (4.4-6) is recognized as the SSP gain. While Eqn.
(4.4-6) reveals some insight into performance, it is simpler
to use Eqn. (4.4-5) for computing P cw/ SSP where P(DI0)
represents the probability of failure (i.e., missing the
target). A failure can arise by having either 1) less than J
partial successes and no full success occurring in L scans
(event D12), or 2) J or more partial successes occurring in
L scans without the occurrence of a full success but the
partial successes do not occur within the K scan deep slid-
175
ing window (event D13). Since event D12 and Dl3 are mutu-
ally exclusive, it follows that
P(DlO) = P(D12) + P(D13).
Note that the probability of event D12 is given by
J-1
P(D12) = ( piqL-i (4.4-8)
To count the number of ways to fail in the second case for
l<J<K, J<K<L, the outcomes in events D11 and D9 are consid-
ered for each J, K, and L as appropriate. The difference in
the number of outcomes in D11 and D9 when there are i>J par-
tial successes is denoted by NL(JK). This difference is the
number of ways to fail when at least J partial successes oc-
cur in L scans without the occurrence of a full success.
For each J and given L, P(Dll) is given by
L LP(D|I) .= ( ) piqL-
This shows, for a given J, that there are exactly
L!/(i!(L-i)!), i = J , -- , L, possible ways for a partial
success to occur in i scans with neither a partial nor a
full success occurring in (L-i) scans. Event D9 is clearly
a subset of event DlI. For i J, the number of elements in
DlI but not in D9, which we have denoted by NLI(J,K), is most
readily counted by forming a table of all possible outcomes
176
for event DlI and applying the K scan deep sliding window to
identify those which do not belong to the event D9. Such
tables for J=2 and K<L=6,- and J=3 and K<L=6 are shown in
Tables 4.4-1 and 4.4-2, respectively. Having determined
N J(JK), the probability of event D13 is
L
P(D13) f E N(J' K)piqL1 (4.4-9)
An example illustrates the procedure for computing
Pclw/ SSP when J=3 and K=L=6. From Eqn. (4.4-8), the number
of ways to fail when i<3 is L!/[i!(L-i)!], i = 0, 1, 2. The
corresponding probabilities of failure for these values of i
are
i-l: 6pq5
i --Z. 15p2q4
From Table 4.4-2, the number of ways to fail when i=3 and
i=4 are:
i=3: 0
i=4: 0.
It is obvious that the number of ways to fail when partial
successes occur in 5 or more scans is zero. Hence, P(DlO) =
P(Dll) for i>5. Using the above results in Eqn. (4.4-5)
177
Table 4.4-1 Vaws to Fail in J=2 of sL=6 SSP
Combinations of 1.2 Partial Successes and 4 Failures over L=6
Seems (15 possible ways for the paq 4 teruj
Scan No.1 11111 0 0 0 0 0 0 0.0 0 02 1 0000 1 1 1 1 0000003 0 1 0 0 0 1 0 0 0 1 1 1 0 0 04 0 0 1 0 0 0 1 0 0 1 0 0 1 1 05 0 0 0 1 0 0 0 1 0 0 1 0 1 0 16 0 0 0 0 1 0 0 0 1 0 0 1 0 11 A (2,K)
Scan Number of WaysDepth (K) to Fail
2 x x x x x x x x x x 103 x x x x x x 64 x x x 35 x 1
6 or more 0
Combinations of 1.3 Partial Successes and 3 Failures over Ln6
Scans 120 possible ways for the pq3 tru)
Scan No.1 1111111111 0 0 0 0 0 0 0 0 0 02 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 03 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 04 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 15 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 16 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1 N 3(K
6 0 1 10 0 0Scan Number of Ways
Depth (K) to Fail2 x x x x 4
3 or more 0
178
-7m-
Table 4.4-2 Ways to Fail In J=3 of M=6 SSP
Combinations ot l.3 Partial Successes and 3 Failures over L=6
Scan 20 possible ways for the p~q3 term)
Scan No.1 I I I I I I I I I 1 0 0 0 0 0 0 0 0 0 02 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 03 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 04 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 15 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 16 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 l1-6(3K)
Scan Number of WaysDepth (K) to Fail
3 x x xx x x x x x x X x x xx x x 164 x x x x x x x x x x 105 x x x x 4
6 of more 0
Combinations of 1.4 Partial Successes and 2 over Lw6 Scans
US possible ways for the pVq2 term)
Scan No.1 I I I I I I I I I 1 0 0 0 0 02 1 1 1 1 1 1 0 0 0 0 1 1 1 1 03 1 1 1 0 0 0 1 1 1 0 1 1 1 0 14 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1S 0 1 0 1 0 1 1 0 I 1 1 0 1 1 16. 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 N4(3,K)
Scan Number of WaysDepth(K) to Fail
3 x x x xx x 64 x 1
5 or more 0
179
yields
PC ,/ P = I - P(DIO) = I - (q646pq5+15p 2q4 ) . (4.4-10)
The alternate form of Pcjw/ SSP by Eqn. (4.4-6) is
PC [w/ SSP P(D8) + P(D9) = I - (l-P)L + P(D9)
= 1 - (I - P)6 + [ 20p3 q3+15p 4q2+6p~q4p6/]/ " (4.4-11)
That the above two expressions yield the same numerical re-
sult is illustrated in Table 4.4-3. In this example, it is
assumed that Pdl = Pd2 for simplicity such that the detec-
tion probabilities in all 3 slants are identical. This is
equivalent to using one average beam shape loss for each of
the 3 slants. The result of the example is of course ex-
pected. That Eqns. (4.4-10) and (4.4-11) are identical can
be shown by substituting (p+q) for P in Eqn. (4.4-10) and
carrying out the algebra:
PCjW/ ssP M P(D8) + P(D9) = 1 - (I-p)L + P(D9)
= 1- E p6 46p'q+15p4q2+2op3qS+15p 2q4+6pq4q]
+ [ 20p3q3+15p 4q2+6p5q+p6]
- I - (q6+6pq5+15p 2q4) .
Following the same procedure, the expressions for
Pc w/ SSP for J=2 and for J=3, and k<L=6 are determined.
180
Table 4.4-3
An Illustration for Computing P IV/ SSP by Two Formulas
(3-Slant Configuration with 6-Scan Deep SSP)
p P(D8) P iw/ SSP P 1w/ SSP
Egn.4.4-5 Ean.4.4-6
0.1177031 0.0664269 0.0902742 0.69027440.1406250 0.0901633 0.1280342 0.12803420.1644844 0.1184709 0.1744983 0.17449840.1890000 0.1514509 0.2294627 0.22946280.2138907 0.1890810 0.2920931 0.29209300.2388750 0.2312025 0.3609468 0.36094680.2636719 0.2775120 0.4340711 0.43407110.2880000 0.3275578 0.5091608 0.50916100.3115782 0.3807418 0.5837611 0.58376100.3341251 0.4363300 0.6554816 0.65548160.3553594 0.4934678 0.7222001 0.72220000.3750000 0.5512050 0.7822266 0.78222680.3927657 0.6085276 0.8344148 0.83441500.4083750 0.6643983 0.8782052 0.87820530.4215469 0.7177992 0.9136057 0.91360570.4320000 0.7677816 0.9411144 0.94111420.4394531 0.8135140 0.9616017 0.96160160.4436250 0.8543281 0.9761709 0.97617090.4442344 0.8897591 0.9860174 0.98601750.4410001 0.9195747 0.9923041 0.9923042
-proQram used for computation:
10 P1 - .220 PRINT " P PC PCi PC2"30 FOR I = 1 To 2040 P1 = P1 + .02550 PP - P1^360 p = 3*P1^2*(1 - P1)70 Q= 1 -P -PP80 PCi = 1 - (Q6 + 6*P*QA5 + 15*PA2*QA4)90 PC = 1- (1 - PP)^6100 PC2 = PC + 20*PA3*QA3 + 15*PA4*QA2 + 6*PA5*Q + P 6110 PRINT USING " 1.####f##"; P, PC, PCI, PC2120 NEXT
181
The results are listed in Table 4.4-4. Using these formulas
and Eqn. (4.4-4), cumulative detection probabilities as a
function of scan depth K for the 5 target models are com-
puted by numerical integration. The results are shown in
Table 4.4-5 (a) through (f) for Swerling cases 1 through 4
and Marcum target models.
Recognizing that the baseline configuration corresponds to
a scan depth of unity, it is evident from Table 4.4-5 that a
only a modest improvement is achieved by SSP. In an attempt
to explain more optimistic projections by other investiga-
tors, both the interim detection probabilities arising from 2
of 3 post detection integration before SSP and the hybrid de-
tection probabilities after SSP, defined by Eqn. (4.4-2) are
computed. The former is given by
Pd(3/3) = P(DI) + P(D2)
while the latter is given by
Pd(hyb) = P(D4) + P(D7).
Values of J=3 and K=8 were used for evaluating Pd(hyb) in or-
der to get more optimistic results. Numerical results for
these quantities are shown in Table 4.4-6 (a) through (e) for
Swerling cases 1 through 4 and Marcum target models. Cumula-
tive detection probabilities under the scan limit of L=6 for
both with and without SSP are also shown. These results are
182
C%4-aC ~
aL_-
LO 0) C
+ +1
C~JC~JCN C'14
,0C-I cl c-I i-.0+ + +0 LA L LA LA
u a aa a a 0
4- ' -%LO LO LO LOo00. 0.L 92- + +
4J f rI -i c C7 a C7a
ccI u- c -I r - -0) en I
J64 ra
++Lucr ay ar Y-
.r4 CZ c CJ alclm u- t OWLOL
tn U) , + + + + +
In W. 1. .0 LO to0
Xj
CL a
c -4 C%j ~. LA to0
183
Table 4.4-S. Cumulative Probability of Detection vs. SNR, 3-Slant,Normal Time Overhead, as a Function of Scan Depth Used
a. Swerling Case 1
c W/ hssp In 6 scans for 2 of X SSP Pc w/ sspin 6 scans for 3 of K SSP-oN scan depth used scan depth used
(d) Pd(3/3).l 2 3 4 5 S 2 3 4 S
10.6 .163 .66 .71 .74 .7S .77 .77 .6i .66 .67 .67 .58 .6319.2 .17S .69 .74 .76 .78 .79 .80 .63 .69 .69 .70 .71 .7210.4 .188 .71 .76 .79 .80 .81 .32 .71 .71 .72 .73 .74 .7510.6 .201 .74 .79 .81 .83 .83 .84 .74 .74 .75 .76 .77 .7710.1 .215 .77 .81 .83 .9S .85 .5i .77 .77 .M7 .78 .79 .8011.9 .229 .79 .83 .85 .86 .87 .69 .79 .79 .80 .81 .81 .8211.2 .243 .81 .85 .87 .88 .89 .89 .81 .81 .82 .83 .83 .8411.4 .257 .83 .87 .89 .S0 .90 .91 .83 .83 .84 .85 .A5 .8611.6 .272 .85 .89 .90 .91 . 2 .92 .As .85 .86 .87 .97 .0811.0 .297 .87 .90 .91 .92 .93 .93 .87 .87 .88 .88 .89 .89I'.6 .302 .99 .91 .93 .S3 .94 .94 .98 .88 .89 .90 .90 .91
12.2 .310 .9 93 .94 .94 .9s .35 .90 .90 .91 .91 .92 .V212.4, .333 .91 .94 .95 .95 .96 .35 .91 .91 .92 .92 .93 .9312.6 .349 .92 .95 .95 .96 .96 .96 .92 .92 .93 .93 .94 .9412.8 .36S .93 .95 .96 .97 .97 .97 .93 .93 .94 .94 .95 .9513.6 .381 .94 .96 .97 .97 .97 .97 .94 .94 .95 .95 .9 .6
b. Swerling Case 2
Pc w/ ssp in 6 scans for 2 of KSSp Pc w/ ssp in 6 scans for 3 of KSSP
So/N scan depth used scan depth used(dB)Pd(3/3) 1 2 3 4 s 6 1 2 3 4 5 6
10.6 .037 .2" .39 .47 .sZ .s .57 .N0 .29 .24 .28 .31 .. 316.z .642 .23 .43 .S1 .Ss .S9 .61 .23 .23 .27 .32 .35 .3-16.4 .848 .2S .47 .Ss .60 .63 .6S .25 .25 .31 .35 .39 .4210.6 .054 .28 .S1 .60 .64 .61 .69 .28 .28 .34 .39 .43 .4610.8 .so .31 .55 .64 .68 .71 .73 .31 .31 .3- .43 .49 .5011.6 .067 .34 .59 .68 .72 .75 .7i .34 .34 .41 .47 .52 .5511.2 .075 .37 .63 .71 .7S .78 .86 .37 .37 .4S .S1 .56 .5911.4 .683 .49 .67 .75 .79 .81 .93 .40 .40 .49 .5 .60 .6311.6 .A1 .44 .70 .79 .82 .84 .9 .44 .44 .M .S9 .i4 .6111.9 .160 .47 .74 .91 .94 .87 .8 .47 .47 .Si .63 .6a .7112.6 .110 .50 .77 .84 .87 .89 .90 .50 .SO .60 .67 .72 .7512.2 .120 .54 .9 86 .89 .91 .92 .SA .S4 .64 .71 .75 .7812.4 .131 .S7 .03 .88 .91 .92 .93 .57 .S7 .67 .74 .79 .8112.6 .142 .66 .9s .90 .92 .94 .94 .69 .60 .71 .77 .92 .9412.8 .1S4 .63 .87 .92 .94 .95 .96 .63 .63 .74 .80 .84 .8713.6 .166 .66 .89 .93 .95 .96 .96 .66 .66 .77 .93 .97 .59
184
Table 4.4-5 Cumulative Probability of Detection vs. SNR, 3-Slant,Normal Time Overhead, as a Function of Scan Depth Used
(Continued)
c. Swerling tase 3.
Pc w/ ssp in 6 scans for 2 of K SSP PQB / sspin 6 scans for 3 of K SSP
f/N scan depth used scan depth used(dB)Pd(3/3) 1 2 3 4 S 6 1 2 3 4 5 6
1.9 .139 .59 .69 .73 .75 .77 .78 .S9 .S9 .61 .63 .6s .6613. .153 .63 .72 .76 .79 .8 .81 .63 .63 .As .67 .69 .7811.4 .198 .97 .76 .88 .82 .83 .84 .67 .67 .69 .71 .73 .7416.6 .184 .71 .79 .93 .84 .86 .96 .71 .71 .73 .75 .76 .77
16.3 .201 .74 .82 .As .87 .99 .9 .74 .74 .76 .78 .79 .9011.6 .219 .77 .es .88 .9 9 . .1 .77 .77 .79 .81 .82 .8311.2 .237 .89 .97 .99 .91 .92 .82 .8 .88 .82 .84 .As .as11.4 .256 .83 .89 .92 .93 .93 .94 .83 .3 .As .9 .97 .9811.6 .27S .86 .91 .93 .94 .95 i5 .96 .96 .97 .e9 .90 .So11.9 .295 .88 .93 .94 .95 .96 .96 .98 .89 .89 .91 .91 .9212.6 .316 .96 .94 .96 .96 .97 .97 .93 .96 .9l .S2 .S3 .9412.2 .337 .92 .95 .96 .97 .97 .98 .92 .92 .93 .94 .94 .9512.4 .3S8 .93 .96 .97 .98 .98 .98 .93 .93 .A4 .95 .96 .9612.6 .380 .94 .97 .99 .99 .9 .99 .94 94 As .S A9 .9712.0 .462 .95 .98 .9 9 .99 .9 .95 .95 .96 .S7 .97 .9713.6 .424 .96 .99 A9 .99 .99 .99 .96 .96 .97 .98 .AS .98
d. Swerling Case 4
Pc w/ ssp in 6 scans for 2 of K SSP Pc w/ sSpin 6 scans for 3 of K SSPSo/N scan depth used scan depth used(dB) Pd(3/3) 1 2 3 4 S 6 1 2 3 4 S 6
10.. 04S .46 .S5 .68 .63 .5 .24 .24 .30 .34 .38 .4116.2 .053 .29 .S .60 .6S .69 .70 .29 .28 .34 .39 .44 .4616.4 .061 .31 .57 .65 .70 .73 .74 .31 .31 .38 .44 .49 .A,16.6 .670 .35 .62 .70 .75 .77 .79 .35 .35 .43 .So .55 .so14.8 .903 .39 .67 .7S .79 .81 .83 .39 .39 .48 .SS .60 .6311.6 .691 .44 .71 .79 .83 .85 .86 .44 .44 .S3 .6 .65 .6911.2 .103 .48 .76 .93 .96 .98 .89 .48 .49 .58 .65 .7 .7311.4 .116 .52 .90 .86 .99 .91 .92 .52 .52 .63 .70 .75 .7911.6 .130 .57 .83 .89 .91 .93 .94 .S7 .57 .68 .75 .79 .8211.9 .146 .61 .86 .Al .93 .A5 .As .61 .61 .72 .79 .83 .8612.6 .162 .A5 .89 .93 .95 .96 .96 .AS .6S .77 .93 .86 .8912.2 .179 .69 .91 .95 .96 .97 .97 .69 .69 .89 .96 .89 .9112.4 .197 .73 .93 .96 .97 .99 .99 .73 .73 .94 .89 .92 .9312.6 .217 .77 .9 .97 .98 .99 .9 .77 .77 .87 .91 .94 .9s12.9 .237 .90 .96 .98 .99 .99 .99 .80 .80 .90 .93 .9S .9613.0 .257 .83 .97 .99 .99 .99 .99 .83 .93 .2 .95 .S7 .97
185
Table 4.4-5 Cumulative Probability of Detection vs. SNR, 3-Slant,Normal Time Overhead, as a Function of Scan Depth Used
(Continued)
e. Marcum (Non-Fluctuating Case)
Pc w/ ssp In 6 scans for 2 of K SSP Pc w/ ssp in 6 scans for 3 ofK SSP
9'o1N scan depth used scan depth used(dB) Pd(3/3) I 2 3 4 5 6 I 2 3 4 5 S
13.3 .iS2 .28 .54 .64 .69 .72 .73 .28 .28 .35 .42 .47 .5112.2 .364 .33 .62 *.71 .75 .78 .86 .33 .33 .42 .49 .5 .5814.4 .379 .39 .69 .78 .82 .84 .85 .39 .39 .SO .57 .63 .5613.6 .As .4S .78 .83 .87 .89 .90 .45 .45 .57 .65 .71 .7413.8 .115 .52 .82 .88 .91 .92 .93 .S2 .S2 .65 .73 .78 .8111.9 .137 .59 .87 .92 .94 .95 .96 .s9 .59 .72 .79 .84 .86I 1.2" .162 .65 .91 .95 .96 .97 .98 .S .65 .79 .85 .89 .9111.4 .191 .72 .94 .97 .S8 .98 .99 .72 .72 .85 .98 .93 .9411.6 .223 .78 .96 .98 .99 .99 .99 .78 .78 .89 .93 .96 .9711.8 .258 .83 .98 .99 .99 1.31 1.63 .83 .83 .93 .96 .97 .9812.6 .296 .88 .99 1.6 1.36 1.0 1.80 .88 .88 .96 .98 .99 .9912.2 .336 .91 .99 1.9 1.03 1.00 1.00 .91 .91 .97 .99 .99 1.0912.4 .379 .94 1.63 1.00 1.00 1.00 1.06 .94 .94 .99 .99 1.03 1.0012.6 .425 .AS 1.30 1.360 1.313 1.60 1.03 .96 .98 .99 1.06 I ' 1.3012.8 .471 .98 1.33 1.33 1.00 1.66 1%0 .98 .98 1.90 1.36 1.00 L.O013.0 .519 .99 1.3, 1.36 1.33 1.03 1.1o .99 .99 1.00 2.88 1.00 1.18
186
Table 4.4-6
Probability of Detection vs. SNR, 3-Slant,
Normal Time Overhead
Pfa Pf rT(dB) LBS1 (dB) LBS1 (dB)
0.339xI0 - 9 0.760xI0-3 8.5626 0.296 3.429
a. Swerling Case 1 PC PCPd Pd- Pd wo w
S/ (3/3) (2/3) (hyb) ssp ssp
10.00 .16 .33 .28 .6s .6910.20 .18 .34 .30 .69 .721e.40 .19 -3s .32 .71 .7510.S .20 .37 .34 .74 .7710.80 .21 .39 .3s .77 .8011.00 .23 .40 .8 .7S .S11.20 .24 .4: .40 .81 .8411.40 .n6 .44 .41 .83 .8611.60 .27 .45 .43 .95 .8611.80 .29 .47 .45 .87 .AS12.00 .30 .48 .47 .8s .!I12.20 .32 .50 .49 .90 .9712.40 .33 .5i .50 .91 .9332.60 .35 .53 .62 .9 .94
12.80 .36 .54 .53 .93 .9513.00 .38 .56 .53 .94 .96
b. Swerling Case 2 Pcw/o
(B (3/3) (2/3)(hyb) ssp ssp
1o:0o .04 .Z7 .18 .20 -.3310.20 .04 .29 .21 .23 .3710.40 .As .31 .L3 .2s .:10.S .as .33 .:6 .28 .4610.90 .06 .3s .29 .31 .S11.00 .07 .37 .31 .34 .SS11.20 .0 7 .40 .34 .37 .5311.40 .08 .42 .37 .40 .6311.S0 .09 .44 .46 .44 .s11.80 .10 .4S .43 .47 .7112.00 .11 .4s .45 .sa .7S1!.20 .12 .SO .48 .54 .7812.40 .13 .s .s1 .S7 .8132.60 .14 .SS .s3 .60 .8412.80 .Is .57 .Ss .63 .8713.00 .17 .S9 .sa .66 .89
- 137
Table 4.4-6
Probability of Detection vs. SNR, 3-Slant,
Normal Time Overhead (cor.t'd)
c. Swerling Case.3
Sro/N Pc PcLA) Pd(3/3) Pd(2/3) Pd(hyb) w/ossp w/ ssp10.00 .13901 .34330 .5998 .59S4 .SS82110.Z .15332 .37020 .31661 .63159 .6978s10.40 .1684S .3913S .3S342 .66341 .73s5710.60 .18444 .41Z68 .38 11 .70S73 .7704310.80 .20120 .43411 .40634 .74021 .802761l.00 .21873 *4SSSa .4326 .772S9 .8321011.20 .23697 .47702 .45805 .80264 .886411.40 .2S588 .49836 .48294 .830Z2 .8820911.60 .27540 .S19S4 50718 .85325 .9026311.80 .Z9S47 .5405 .-3072 .87770 .9207912.00 .31602 .6118 .E354 .89761 .93Ss12.20 .33700 .581 3 .57563 .91507 .9483412.40 .35832 .60149 ,Sa701 .93019 .950012.60 .37992 .62103 .6176 .94315 .9677912.60 .40171 .64010 .63760 .95414 .9749Z13.30 .42363 .65867 .5683 .96234 .9865S
d. Swerling Case 4
So/N Pc PC(d.B Pd(3/3) Pd(2/3)Pd(hyb) w/ossp w/ ssp10.00 .04S10 .31140 .25ee .24329 .4076510.20 .85270 .33725 .26435 .27735 .4s:6410.40 .06087 .36390 .29999 .31397 .5189010.60 .06997 .39120 .33646 .35287 .5752610.90 .08003 .41902 .7321 .3S37 .630S:11.00 .09108 .44721 .40979 .43617 .6235311.20 .10317 .47562 .44578 .47965 .7332711.40 .I1629 .50409 .46088 .52374 .7791911.6 .13048 .S3247 .51485 .56779 .8198611.80 .14571 .56060 ,54757 .61127 .8557812.00 .16197 .58834 .57894 .65361 .996S812.20 .17924 .61554 .60893 .69430 .9124012.40 .19746 .64208 .63755 .73297 .9335712.60 .21665 .66784 .65481 .76893 ,8505412.80 .23668 .69273 69075 .90219 .9638513.00 .25750 .71663 .71537 .83243 .97406
188
Table 4.4-6
Probability of Detection vs. SNR, 3-Slant,
Normal Time Overhead (cont'd)
e. Marcum (Non-Fluctuating Case)
So/N Pd Pd Pd P'o PC(dB) (3/3) (2/3) (hb.) ssp .ssp10.00 .0S .36 ;29 .25 .5310.20 .A6 .40 .35 .33 .Sa10.40 .08 .44 .40 .39 .6i10.60 .10 .49 .46 .45 .7410.80 .11 .s53 .51 .S .8111.00 , 14 . .. 7 .= .8,.20 .16 .6: .6i .63 .S11.40 .19 .69 .66 .7: .94
11.60 .22 .71 .70 .78 .97I1.80 .26 .75 .74 .A- .9612.00 .30 .78 .78 .88 .9912.20 .34 .8: .62 .91 1.0012.40 .3s .85 .As .94 1.0012.50 .42 .86 .98 .96 1.001:.S0 .47 .90 .80 .Sj 1.001 .00 .52 .92 .9. .9 1.00
189
also plotted in Figures 4.4-1 through 4.4-5. The measure of
improvement can be obtained by drawing a horizontal line
passing through the ordinate at 0.9 and comparing the differ-
ence in (S0/N) for cumulative detection probabilities with
and without SSP for each of the models. The improvement mea-
sured this way is 0.2 dB, 2.0 dB, and 0.9 dB for Swerling
case 1, case 2, and Marcum target models, respectively. A
larger improvement can be claimed if a similar comparison is
made between Pd(3/3) and Pd(2/3) or Pd(hyb) at a level of
0.32 probability or at any other level. Even a larger im-
provement can be claimed if the cumulative detection prob-
ability with SSP is compared against Pd(3/3).
A set of computations were also made using the reduced
time overhead. The result for Swerling case 1 is given in
Table 4.4-7. The difference in detection performance be-
tween the two cumulative detection probabilities, one with
and the other without SSP, is almost identical to that based
on the normal time overhead as expected.
190
E-4 0
0 -
P-4-
1*
I I-
II
C
0 ; C; C; C;C' C; '-4 0
uonzD;I~a jo SITIq-eqoia
191
E-43IL A II x I A% AAA.
IN a A - -
a A I -I -IN A 09
A A A 14 %4) -
PIAN C)
o0 A A 4
0 mC) 0
eli ba 04 $-
o IIV
I al
C4
0 0 0 0) 0) 0) 0 0 0 0 00)0) Go r-, to I) ~- n~ N~ V-0
uofolzap jo Kj~jiqeqo-,a
192
off # I I 4a I a4r
all a a I t
4J
a ty
0 0
4, to>
4L 4 L
0 ~ ~ ~ ~ r 0i N- IS DL)~r~cJ~
1934
01 f
t-A
1 1 1I I U r, t
o z
1 11 V
>1
- .0
01 0t 4
t U-$4
II -- -
144
0 ; C; C C; at) C; P' C;I 0*
uogoa jo SITflqeqoJic
194
E-1 0
01
4VI
~mI/2 >11111I
zz
bDo*
LA.2
9Lm, :U-r
C;C
uorpoa-4Q jo 91111qqoial
195
Table 4.4-7 Probability of Detection vs. SNR, 3-Slant,
Reduced Time Overhead, Swerling Case 1
-9-0IN PC PCJdE) Pd(3/3) Pd(2/3J P4Whb)w/O Asp w/ ssp
9.50 .08526 .21392 .14460 .41414 .43747
8.70 .0937S .22749 .16027 .44602 .47107
8.90 .10276 .24140 .17671 -.47827 .50490
9.10 .11229 .25562 .19384 .51066 .53865
S.30 .12234 .27012 .21)57 .54296 .S7213
9.50 .13290 .28487 .22990 S574S7 .60504
S.70 .14396 .29984 .24843 .603647 .63716
9.90 I1SSSO .31500 .26738 .6372S .66828
10.10 Tr67S0 .33032 .29653 .6671" .6e'970
10.30 .17996 .34577 .30S79 .69591 .72676
10.50 .19285 .36132 .32509 .72348 .7S393
10.70 .20614 .37693 .34434 .74971 .77929
10.90 .21982 .39258 .36346 .77449 .8030e
11.10 .2338S .40823 .38240 .79775 .82515
11.30 .24819 .4:6387 .40111 .81944 .8450
11.50 .26294 .43946 .41953 .83954A .86412-
ti.70 .27775 .4S497 .4376S .95805 .8810
11.90 .292-99 .47039 .4AS S44 A7499 .89526
12.10 . 3 0822 .48568 .A74182 .89040 .91013
12.30 .32373 .50083 .48986 .90434 S"&:37
12.S0 .33937 S515al S506513 .91687 .933:6
11.70 .35511 .53061 .5276 .92907 .942971
1S.e .37092 .54521 SUEZ6 .93802 .95)3-0
13.1 .38677 -S5953 SZ-409 .94682 .95966
13.30 .40253 S57373 .5S317 SS4SS .96504
13.50 .41847 .08763 .S8387 .9513-& .970SS
13.70 .43426 . 60 1 27 .59918 .9672 1 .97530
13.90 .44997 .51465 .6 1212 .97231 .97935
14.10 .4G5S8 .6Z774 .625S69 .97670 .98291
14.3e .498107 .64055 .63899 .99047 .995T3
14.50 .40641 .65308 G61730 .98369 .92620
14.70 .51158 .66530 .66422 .99642 .990:7
14.90 S52657 .5?772 .67636 S98874 .99:00
15.10 .54134 .6as8A .68816 .99069 .9534
15.30 .53590 .70016 .6B96) .992t33 .9946S
15.50 S57e2l .71116 .71073 .99370 .99564
15.70 .58429 .712187 .72153 .99484 .99646
15.90 .598 .7322S .73199 .99578 .99714
16.10 .61160 .741235 .74214 .99657 .99769
16.30 .62485 .75214 .75198 .99721 .99814
16.50 .63780 .76163 .76150 .99774 .99850
16.70 .65045 .770182 .77072 .99818 .99880
16.90 .66291 .77972 .77964 .99853 . 99904
17.10Q .67486 .7863"2 .7827 .99821 .99923
17.30 .68659 .79665 .79661 .99905 .999--s
17.50 .6980'" .80470 .90466 .99924 .99951
196
4.5 Detection Performance under Conventional J of K SSP
Finally, detection performance using conventional J of K
SSP is examined where K equals the number of scans L. This
is based on 2-slant, 2 of 2 post detection integration for
the reasons stated in Section 4.3. In addition to the
events introduced in Sections 4.3 and 4.4, the following two
are defined:
D14: detections occur in both of the 2 slants in the 2-slant configuration
D15: event D14 occurs in J or more of the K=L scans
The single scan detection probability P(D14) is given by
P(DI4) - Pd(2/2) 2 (4.-1(4.5-1)
where Pdl denotes the single slant detection probability and
the method for averaging with respect to target RCS fluc-
tuation is as described in Chapter 3.
The cumulative detection probability is
Pei w/ P = P(D15)K (K~p
E p Pd) K (4.5-2)
Results for the single scan detection probabilities P(D14),
197
obtained by numerical integration, are listed in Table 4.5-1
and are plotted in Figure 4.5-1 for the 5 target models.
The cumulative detection probabilities P(D15) for J=2 and
J=3 with K-6 are shown in Table 4.5-2 for the same target
models. They are plotted in Figures 4.5-2 and 4.5-3 for J=2
and J=3, respectively. Recall from Section 4.3 that the
choice J=2 and K-6 results in increased false alarms beyond
an acceptable level. Therefore, comparison with the
baseline configuration is made for the case where J=3.
The above results are based on using the reduced time
overhead. Hence, they should be compared with the baseline
results which are also based on the reduced time overhead.
Because the integrated signal-to-noise ratio is configura-
tion dependent (i.e., depends on the number of slants in a
beam dwell), it is necessary to introduce a reference range
so that a common reference point in signal-to-noise ratio
can be established for different configurations. The refer-
ence SNR for the baseline is 12.2 dB. This means that the
received So/N from the reference target located at the ref-
erence range is 12.2 dB for the baseline. When the reduced
time overhead is assumed in the baseline, the number of
pulses for integration increases such that the received S0/N
from the reference target at the reference range increases
to 13.5 dB. In the 2-slant configuration, SNR per slant is
198
Table 4.5-1 Single Scan Probability of Detection vs. SNR,
2-Slant with a Reduced Time Overhead
Pfa -6 Pf . 3fl N rT LBS1.197x0 1.-o94X10 2 2 8.Z74 2.S28
So/N (dB) SUl SW2 SW3 SU4 MARCUM------------ -------------11.00 .311 .177 .329 .222 .31511.20 .326 .189 .348 .248 .35411.40 .342 .202 .369 .260 .39511.60 .358 .215 .391 .280 .43811.80 .373 .229 .412 .30 .48312.00 .389 .243 .434 .322 .52912.20 .405 .258 .456 .343 .57S12.40 .420 .272 .477 .365 .62112.60 .436 .287 .499 .389 .66612.80 .452 .302 .528 .410 .71013.00 .4S7 .318 .542 .433 .75113.20 .483 .333 .5G2 .456 .79013.40 .498 .349 .583 .478 .82513.60 .513 .364 .603 .501 .95713.80 .528 .380 .623 .523 .88514.00 .543 .396 .642 .545 .90914.20 .557 .412 .661 .567 .93014.40 .571 .427 .679 .58 .94714.60 .585 .443 .696 .609 .96114.80 .599 .459 .713 .630 .97115.00 .612 .474 .730 .649 .980IS.20 .626 .489 .746 .669 1.00015.40 .638 .50O .761 .67 1.00015.60 .651 .520 .775 .705 1.80015.8 .663 .535 .789 .722 1.80016.00 .675 .549 .802 .739 1.00016.20 .687 .564 .815 .755 1.00016.40 .699 .578 .827 .770 1.0016.60 .710 .592 .838 .785 1.0016.80 .728 .605 .949 .799 1.0017.00 .731 .619 .959 .812 1.0017.29 .741 .632 .968 .824 1.80017.40 .751 .644 .877 .836 1.80
17.68 .760 .657 .986 .847 1.0017.980 .770 .S69 .894 .58 1.0018.00 .779 .681 .981 .868 1.0018.20 .787 .693 .909 .877 1.00018.40 .796 .704 .915 .96 1.0018.60 .804 .71S .921 .894 1.0018.80 .811 .725 .927 .902 1.00019.0 .819 .736 .933 .909 1.88819.20 .826 .746 .938 .916 1.00019.40 .833 .755 .942 .922 1.000
199
IaFIE V 1-4
PW4.
E-4.
oc -'.4 $4
'.4
be *
cn CM
ON
-'.4
0 m~ w r-, wo o -- ) C~
uiai jo qoaiJ T1eDS aI~uiS
200
Table 4.5-2 CUMULATIVE PROBABILITY OF DETECTION VS. SNR. 2-SLANT,PULSE NORMAL WITH A REDUCED TIME OVERHEAD
2 of 6 SS 3 of 6 SSP(dB) .5w.2 SW3.. -SW4 MOJM SWi . SW2 S43 5W4 MARCW4
C.O .1492 .09 -6976 .62S8 .0141 .6244 .6619 .6122 .915t .Sees0.26 .179S .6346 .1162 .6329 .6196 .6304 .6624 .6163 .0622 .00101.40 .1934 .6410 .1306 .6418 .6269 .6372 .6033 .6226 .6033 .66160.66 .2199 .GS82 .1643 .G517 .0370 .94S7 .0943 .0295 .004S .60288.80 .2472 .0608 .1934 e065s .6517* .0560 .9858 .0372 .0068 .00459.60 .2761 .6722 .22S9 .009 .68S '.967S .6676 .0482 .691~ .00719.20 .3054 east1 .2592 .9999 .9915 .9902 .6191 .9607 .0126 .01119.4" .3373 .1009 .2956 .1262 .1292 .9S2 .9138 .9759 .0172 .017"t2.60 .3692 .1182 .3348 .1429 .156 .Ills .0167 .6946 .0227 .92S]9.80 .4035 .1365 .3766 .1799 .2093; .1369 .6211 .1156 .0364 .039S16.866 .4359 .1579 .4205 .2326 .2529. .1494 .6267 .1406 .9402 .657910.20 .4768 .1799 .4637 .2353 .3127 .1726 .0330 f1679 .0516 .083619.40 .5636 .2049 .5692 .2713 .3915 .1955 .0419 .1989 .0655 .118"10.60. .5379 .2306 .5513 .3127 .4542 .2213 .0499 .2329 .936 .161619.90 .5711 .2S68 .594S .3545 .5S334 .2483 .0597 .2698 .1939 .2179
11."0 .6033 .2993 .6392 .3986 .6116 .2763 .6727 .3091 .1277 .283911.26 .6342 .3176 .6772 .4422 .6994 .3952 .99 .3489 .1539 .36i011.40 .6667 .3496 .7194 .4996 .7S89 .3368 .1014 .3915 Jes56 .445311.60 .9957 .3915 .752S S53SS .9212 .3691 .1182 .4370 .2196 S534111.90 .7224 .4159 .7849 .S799 .6739 .39S8 .1377 .4996 .2557 .623912.00O .7492 .4494 .8259 .6261 .9i55 .4329 .155 .5259 .2974 .709912.20 .7744 .4849 .8437 .6676 .9463 .4669 .1823 .5766 .3389 .794512.48 .7965 .5174 .0675 .7093 .9679 .4971 .2957 .6122 .3834 SASS512.60. .810S .5513 .9997 .7476 .9920 .530 .2326 .6544 .4399 .899212.80 .9399 S5S41 .9093 .7919 .9907 .5625 .2S94 .6936 .4764 S93--13.90 .9565 .6179 S9252 S9I4S .99S4 .5925 .29S7 .7313 .5239 .9913.26 .9739 .6402 .9306 .8437 .999O .6239 .3199 .7642 S5796 .979813.46 .3897 .6796 .1566 .686 .9992 .6525 .3509 .796S .6141 .989613.60 .9623 .7965 .9664 .0916 .9997 .6993 .3914 .9249 .6591 .99s;13.60 .9147 .7343 .9607 .9107 .9999 .7072 .4143 .9511 .6993 .997814.00. .9259 .7604 .97S2 .9274 1.0090 .7330 .4"74 .8737 .7364 .999114.26' .9354 .7949 .9907 .9416 1."G99 .7562 .496 .0942 .772_91 .999"?14.46 .9440 .9064 .9850 .9532 )100 .7783 S51is .9116 .9938 .999914.60 .9517 .9276 .9994 .9630 1.9600 .7994 .5443 .92r13 .9336 1.999614.39 .95S6 .9473 .9911 .9712 1.6966 .8t94 .5756 .9394 .9596 1.00015.60 .9643 .8643 .9933 .9774 1.1966 .6370 .6063 .9569e .9915 1.99995S.20 .9698 .8800 9956 .9927 1.9990 .8548 .6354 .9663 .9622 1.9615.40. .9749 .8952 S9963 .9967 1.61969 .9691 .6656 .9679 .91B7 1.0669I 5.go'970 .903 .9972 .9999 1.9996 .9837 .6936 .9741 .9334 1.09915.96 .9912 .9261 .9979 .9923 1.199 .81162 .7194 .9794 S94SS 1.00016.006 .9041 .9301 .9995 .9943 1.9909 .9979 .7431 .9936 .9S63 1.9960
201
0
NW. 0C4)
0egg>
ISO, P-1 zd
Z nn
-
(9 jo Z) uoTja;aa jo S£jijiq'qoicj =no
202
~0
L)4
-T t)
0 0I E
.. 2 I-A
0 0 0 0 0 0 0 0 0 00I-
20 3
higher because of one less slant into which the available
number of pulses must be divided and the reduction of the
number of fixed time overheads from 3 to 2. The net effect
.is that S0/N for the reference target at the reference range
increases to 16.0 dB. To make a performance comparison at a
specific cumulative detection probability level, e.g., at
0.9, the SNR margin for each of the competing configurations
is first determined. The margins are compared and the con-
figuration having the larger positive margin is superior by
an amount equal to the margin difference. Figure 4.5-4 il-
lustrates this procedure using some fictitious numbers.
configuration A configuration B
.o0.9 L-- --- ----------.-ao.9---------- -------0 0
margin 0 marginV 3.5dB 7.0dB
* 0
10 13.5 9.0 16.0
reference SNR reference SNR
Configuration B is superior to configuration A by
marginif . B - margin 1 config. A m 7.0 -3.5 - 3.5 dB
Figure 4.5-4
Illustration of Performance Comparison Method
204
The SNR margin for Pc=0.9 for the baseline using reduced
time overhead is 1.3 dB for a Swerling case 1 target.
Similarly, for the conventional SSP with J=3, K=6, the SNR
margin is 0.1 dB as can be seen from Table 4.5-2. There-
fore, the performance under the conventional 3 of 6 SSP is
inferior to the baseline performance by 1.2 dB!
A totally different conclusion can be drawn if the cumu-
lative detection probability under SSP is compared with the
single scan detection probability of the baseline. In the 3
of 6 SSP, Pc=0.9 was reached at S0N= 15.9 dB for a
Swerling case 1 target giving the margin of 0.1 dB. The
values of S0/N required to reach Pd = 0.9 in a single scan
of the baseline for Swerling case 1 target is 23.0 dB. The
SNR margin is 13.5 less 23.0 dB, or -9.5 dB. The difference
between the two is 9.6 dB in favor of the 3 of 6 SSP. This
form of comparison is made by some investigators. Ob-
viously, this results in an erroneous conclusion.
The correct assessmentlis that there is marginal improve-
ment with SSP. This conclusion confirms Barton's assertion
that the simple single scan approach is preferred even for a
stationary target when the scan depth used for correlation
is less than 6 (4). Scan depth beyond 6 is impractical
in most realistic target environments where targets can pop
1. for the cases examined in this report
205
in and out of terrain masking and move in and out of blind
velocity doppler cells or eclipsed range cells either inad-
vertently or intentionally by clever maneuvers. Target ma-
neuvers at 6-g level at a moderate speed would require a
large correlation window which results in false alarm rates
far greater than the baseline specification if the scan
depth used for correlation is larger than 6 as shown in Sec-
tion 4.3.
206
CHAPTER 5
PERFORMANCE IMPROVEMENT WITH SCAN RATE REDUCTION AND
NONCOHERENT INTEGRATION
The radar range equation was introduced in Section 1.1.2.
When the transmitter power P, wavelength at the radar
ooperating frequency X, antenna gain G, target RCS a, number
of pulses for coherent integration NI and system losses are
specified, the SNR in the coherent processing interval (CPI)
for a target at range R is proportional to the ratio of the
energy received in a single pulse to the noise power density
multiplied by the number of pulses coherently integrated.
This is indicated by Eqn. (1.1-10). Of course, the number of
pulses available for integration in a CPI and the number of
CPI's in a given antenna beam direction within a scan is
proportional to the dwell time of the antenna beam over a
point target. This in turn depends on the frame time TF.
To resolve the range ambiguity in a high PRF radar, mul-
tiple looks with distinct PRF's are required in each beam
dwell in a scan. For the radar under consideration, 3 looks
are required for the bandwidth chosen. Because of the prac-
tical fixed time overhead associated with each slant, it is
not beneficial to form more than 3 slants for post detection
integration within the given scan rate constraint even
207
though, in theory, 8 slants with 3 of 8 processing would
perform better. A performance comparison of 2 of 2 post de-
tection integration in a 2 slant configuration, and 2 of 3
and 3 of 3 processing in a 3 slant configuration showed that
the baseline 3 of 3 post detection integration was a good
choice when time overhead, scan rate constraint and range
resolution requirements were taken into account. An im-
provement on the order of 1 dB in equivalent signal-to-noise
ratio can be associated with 2 of 2 or 2 of 3 processing
when the respective cell false alarm probabilities are ad-
justed to yield the same level of system false alarms. How-
ever, the cost for a 3 fold range ambiguity with these pro-
cessing options must be weighted in the comparison.
Under a careful examination (see Chapter 4), scan-to-scan
processing also failed to show the performance improvement
that was hoped for. When the dwell time specification is
expressed in terms of the solid angle n subtending the
search volume and the frame time TF needed to completely
cover the volume once, Eqn. (1.1-14) shows that the SNR for
a target at range R is proportional to the power-aperture
product independent of the operating frequency. When the
power-aperture product is held fixed, it follows that the
variation of TF for a given a is the only remaining option
for additional performance improvement.
208
In Section 1.1.3, and in the introduction of Chapter 4,
the concept of cumulative detection probability Pc is intro-
duced. Because of the loss associated with target RCS fluc-
tuations, it is explained there that it is more efficient
to achieve one or more detections with high probability by
using several scans rather than a single scan. According to
Barton [4], about 6 scans are optimum for detecting a fluc-
tuating target at Pc - 0.9 if the pulses available in a scan
are noncoherently integrated. DiFranco and Rubin [2] show
that the number is 4 for Swerling case 1 and 2 or 3 for
Swerling case 3 when the available pulses in a scan are co-
herently integrated. These results assume that the change in
target range is negligible during the cumulative detection
period. In this chapter the optimum frame time needed to
maximize the range at which the specified Pc is achieved for
a radar with a uniformly scanning antenna is determined for
a target approaching with a constant radial velocity.
First, the classical works by Mallett and Brennan [6) and
DiFranco and Rubin [2] are examined. Based on their re-
sults, a performance test with a slow scan rate (TF=50
seconds) was tried with the consequence that the range was
extended at which Pc - 0,9. The equivalent sensitivity im-
provement was 7 to 8 dB. It is very significant that the
cumulative detection probability for a low observable target
209
whose RCS is less than the reference target by 7-8 dB can be
detected at the same range as the reference target by in-
creasing the frame time. This is a result of a trade-off
between detection and track update. Their work is then ex-
tended in this chapter for a high PRF radar application
where the frame time affects both the SNR per slant and the
system false alarm rate.
Finally, because of the track update requirement, the ex-
tension of the frame time is limited to twice that of the
baseline. 3 and 6 slant configurations with either the bi-
nary or noncoherent post detection integration are then
analyzed. The performance relative to Pc - 0.9 is determined
for detecting a target within time durations of 60, 20, and
10 seconds. Performance, in terms of track update rate, for
targets both in the outer and inner range with respect to
the reference range is also examined. With noncoherent inte-
gration of 3 slants and slant-to-slant frequency agility,
significant improvement is achieved in terms of P c and track
update rate in the outer range. Compared to the baseline,
the track update rate suffers degradation in the inner
range.
210
5.1 Scan Rate Optimization
The surveillance radar equation relating SNR to the radar
power-aperture product when the surveillance volume 0 and
search frame time TF are specified was presented in Section
1.1.2 as
S - PAA TF (1.1-14)-& 4R 4 kTo NFo L 0
When the target approach velocity VR and the desired cumula-
tive detection probability PC are specified instead of TF,
Eqn. (1.1-14) can be put into a form suitable for scan rate
optimization. The optimization is in the sense of
maximizing the range at which the specified Pc is achieved.
Let Rz denote the range at which (S/N) becomes unity.
Then, setting (S/N) = 1 and R = Rz in Eqn. (1.1-14) and
solving for Rz gives
R PA T PA A T F= 0 kT o NFo L l
Let a denote the range interval traversed in TF by a tar-
get closing in with a constant radial velocity VR. The sce-
nario is shown in Figure 5.1-1.
211
CAUjtj-7edetet0o Zone~
a9'Proahu9tre
Figure 5.1-1
Detection Scenario of an Approaching Target with a constantRadial Velocity
Note that
A =VR T1 . (5.1-2)
Expressing T F in terms of VR and A from Eqn. (5.1-2) and
substituting into Egn. (5.1-1) yields,
212
P A a A (5.1-3)"(4w kTo NFo L V "
Let R, be defined such that R. is expressed in terms of R, and A by
R ARIA. (5.1-4)
In terms of R,, Eqn. (5.1-3) becomes
PA A aRI - 4, kTo N 0o L 0 VR (5.1-5)
R3 represents the fixed set of radar parameters (PA, A, kTo , NF o , L, 0) and
target characteristics (a, VR).
For determining the cumulative detection probability, let
L denote the number of times a target will be illuminated
(i.e., the number of scans) during the period in which it is
to be detected one or more times before it reaches the range
Rm. The first illumination of the target is assumed to occur
in the range ring extending from Rm+(L-1)4 to Rm+L4. Let
Rm+(L-l)a+r denote the range to the target when it is first
illuminated where O<r<&. By Eqn. (5.1-2), the target will
be at the range Rm+ia+r when it is illuminated while in the
range ring extending from Rm+iA to Rm+(i+l)A ; i=0,--,(L-l).
The position of the target during each scan when illuminated
is indicated in Figure 5.1-1.
213
Let Pd(R) denote the probability of detecting the target
in a single scan at range R. The cumulative detection prob-
ability Pc of detecting the target by the final range (Rm+r)
is given by
L-1PC M - 11i [ l-Pd(Rm+r i) ] (5.1-6)
Assuming r is uniformly distributed between (RP, Rm+), the
average cumulative detection probability Pc can be written
as
.11-.fi [ I-Pd(Rm+r+iA)])dr . (5.1-7)
Given that the power-aperture product of the radar is fixed,
for a specified target velocity VR and search angle n, it is
sought to optimize the search frame time TF so as to
maximize R, the range by which the target is detected at
least once with a probability of 0.9.
Mallett and Brennan [6, 10] normalized all distances in
Eqn. (5.1-7) with respect to R1 . For different values of Pc'
they solved Eqn. (5.1-7) by numerical evaluation for all
possible pairs of A/R1 and Rm/Rl while letting L and, hance,
the outer range limit get large enough for each trial such
that Pd(Rm+r+(L-1)A) becomes negligibly small. For a given
214
Pc' the A/R1 which corresponds to the maximum Rm/Ri is the
optimum A/RI . Their result shows, for a given target veloc-
ity, that the larger the detection range the longer the
frame time should be. Eqn. (5.1-4) can be written as
where
- Rl(S/N)Z= •
In general, the range R is related to Rz by
R4 (S/N)z 1- (S/IN) - (S/N)
As a result, Rm can be expressed as
Rj R
where (S/N) is the SNR per scan for a target located at Rm .
Assuming all the pulses in a beam dwell are coherently inte-
grated, Mallett and Brennan solved numerically for the opti-
mum values of 4/R1 and R.R 1 for Pfa = lxlO6 for different
values of Pc" The optimum values for P c 0.9 for a
Swerling case 1 target are
AA-i -0.042
V . 0.155
215
Using these optimum values in Eqn. (5.1-2) gives TF as
T? (A/R)R 1 (A/R,)Rm
= VR(Rm/Rl)
- 0.27R , (5.1-8)VR
Typical values for R. and VR are 200 nautical miles and
1600 knots, respectively. The optimum frame time according
to the above equation is then 121 seconds. This contrasts
with the baseline frame time of 10 seconds.
Because the above results suggest that an improvement is
possible by slowing down the scan rate, many detection per-
formance trade-off analyses are carried out in this investi-
gation using a frame time longer than that of the baseline
and either M of N binary or noncoherent post detection inte-
gration as options. The same fixed time overhead as in the
baseline is used for each slant and the cell false alarm
rate was adjusted so that the system false alarm rate is the
same as in the baseline. One sample run with the frame time
close to 50 seconds and a post detection processing consist-
ing of 6 slants of signals noncoherently integrated is
shown in Figures 5.1-2 and 5.1-3 for the single scan and cu-
mulative detection probabilities, respectively. In this
simulation, VR - Mach 1 was assumed and the maximum range
where the cumulative detection process began was 400 nauti
216
Cm
vi In 0
41C.
C
u 14
4'1
C; 0 cv-U'D
64&
N0±133 40 AiflIWU0~d 3AUYMM V 0
A I
I Ilp m- CaeS se
402
x(~ I 14'
C" Cc a 44'~
o6 U
Cc4
D1.I 0'
8 ~ ~ ~ ~ r U 105 4
N011310 A0 A11IW9Old W=~ MUIS
217
cal miles. The time limit of 1 minute to achieve the cumula-
tive detection probability was ignored in this case. The
cumulative detection probability comparison indicates that a
performance improvement equivalent to an increase of ap-
proximately 7 to 8 dB SNR is achieved by slowing down the
scan rate by a factor of approximately 5. If P., were the
only performance criterion, the longer frame time would be
very beneficial.
DiFranco and Rubin [2] followed a slightly different ap-
proach. For a given Rm, they solved for values of A and L
which minimize the power-aperture product. To mae the solu-
tion to Eqn. (5.1-7) more tractable, they let r = A/2 and
Ri - Rm +4/2 + iA. It follows that Pd(Rm+r+iA) - Pd(Ri) .
They also normalized a such that
d VRTF (5.1-9)
Then, carrying out the integration in Eqn. (5.1-7), Pc can
be approximated as
L-I jL Pd iR,.jI~d(2i+1)])] . (5.1-10)
218
Swerling [2, 13] has shown that the detection probability
for noncoherent integration of an incoherent pulse train
can be approximately expressed as
exp(-x) ; for Swerling case I
P (1+2x) exp(-2x) ; for Swerling case 3
where
x = f(Pf,, NI)f
and - is the expected peak SNR per pulse. Let R2 be the range at which
x = 1 = f(Pf , N1)/(N) 2 . Then, from the radar range equation
and
R) f(Ptf,, Q
Consequently, x can be expressed as
x f(Pf., NI)IIA, (_1 ffi
With the above identity, Pd is approximately given by
219
exR[2(k ; for Swerling case I
Pd 1+2 B(5.1-11)
I[1+2 - )JR exp[- 2 (- .) J ; for Swerling case 3.
DiFranco and Rubin substitute Eqn. (5.1-11) into Eqn.
(5.1-10) to obtain the following expression for Pc
VC-Y ( I- [(1+2 (k) 4 (14d(2i+l) )411-
-exp[-I - (1+d(2i+1) )41) (5.1-12)
where 1-1 and 1-i2 applies to Swerling case I and 3, respectively.
Note that Eqn. (1.1-14) can be rearranged to obtain an
expression for the power-aperture product. Substituting
.f(PN 1 ) (R)4(N) k- , T r A2Rmd and R =Rm
into Eqn. (1.1-14), where kI is a constant, and rearranging
for PAA yields
220
PAA - k(R 2/Rm) 4irRkToN FoLno(2R.d/VR)
= k ( 2irkTO L VaRR2)
2(R2/Rm)" (5.1-13)
where all of the terms in coefficient k 2 are constants that
must be specified for a search radar.
For a specified Rm, DiFranco and Rubin find values of d
and L which satisfy Eqn. (5.1-12) for a specified Pc and
also minimize PAA in Eqn. (5.1-13). Their steps are:
1. For a specified Vc, select values of d and L, L=l, 2,
3, --, and solve Eqn. (5.1-12) for R2/Rm . A plot of R2/Rm,
as a function of d, is generated for each integer value of
L.
2. For each value of L, solve Eqn. (5.1-13) for PAA using
pairs of d and R2/Rm as obtained from step 1. A plot of PAA
as a function of d is then plotted for each value of L.
3. Select that pair of values for L an d which yields the
minimum value of PA A .
Minimizing the power-aperture product for a specified Rm
is equivalent to maximizing Rm for a specified
221
power-aperture product. It follows that the values of L and
d which yield a minimum power-aperture product for a
specified Rm also result in a maximum value of Rm when
that minimum power-aperture is specified. The system frame
time is then given by
2RdTF =-v (5. 1-14)
Their results show that the approximate optimum frametime and the number of scans to achieve the required P are
such that the cumulative detection zone is equal to Rm . In
particular,
LVRT, - R. (5.1-15)
That is, the detection zone is equal to the minimum detec-
tion range. Thus, once the optimum L is known, the optimum
TF follows. From the work of DiFranco and Rubin the optimum
L for different P levels are given in Table 5.1-1.c
Table 5.1-1 Optimum L for the Pc Level SpecifiedPc=0.9 P =0.95 Pc=0.99
Swerling 1 4 5 6
Swerling 3 2 3 4
222
Returning to the previous example where 1c=0.9, Rm=20 0
nautical miles and VR=1600 knots, the optimum TF becomes
Rm (200)(36) 112.5 seconds
Vj TL (1600)(4)
This is almost the same result as obtained earlier using the
Mallett and Brennan approach.
Both the Mallett and Brennan and DiFranco and Rubin teams
assume that there is no radar line of sight limit and,
therefore, L can be as large as necessary. Also, without
stating it, the radar is assumed to be operating in a low
PRF mode so that the time overhead, assuming it is entirely
due to the round trip transit time for the radar wave with
respect to the target at the maximum range of interest,
does not affect the division of the available radar energy
into M number of slants and L number of scans.
In the next section, scan rate optimization is carried
out for a high PRF radar. To effectively utilize the
various slants needed to resolve the range ambiguity associ-
ated with the high PRF radar under consideration, post de-
tection processing, either an M of N binary integration
where M>3 or a noncoherent integration where N>3 is required
in each scan. Unlike the low PRF radar studied by others,
223/224
in a high PR? radar, there is a fixed time overhead associ-
ated with each of the N slants and L scans which includes
one round trip transit time for the entire pulse group in a
slant. This affects in a nonlinear way the SNR and cell
false alarm probability and complicates determination of the
optimum frame time. Imposing a line of sight limitation or
time constraint in achieving the stated Pc further compli-
cates the problem.
225/226
5.2 Scan Rate Optimization for a High PRF Radar
In this Section, scan rate optimization for a high PRF
radar is carried out by extending the work of the Mallett
and Brennan team. The extension is in the form of taking
into account the effect of a fixed time overhead on the co-
herent integration gain and the cell false alarm probability
allocation as a function of antenna scan rate. As the scan
rate decreases, it will be shown that the variation in de-
tection performance due to the fixed time overhead can be
ignored once the coherent integration gain and the cell
false alarm probability allocation in a multi-slant system
are properly assigned taking into account the effect of the
fixed time overhead for the particular post detection inte-
gration chosen at its maximum antenna scan rate assumed.
The procedures for computing integration gain, allocating
the false alarm probability, and translating these to the
single scan detection probability after the post detection
integration are developed in Chapters 2 and 3.
To take into account the effect of the fixed time over-
head, the following reference parameters for the baseline
system are introduced:
TF -frame time in the baseline
227
VRo - 450 knotsVR0
&0 - T oV o •
The optimization is carried out for the 3 of 3 post detec-
tion integration scheme as used in the baseline. Let TH
denote the fixed time overhead. In the baseline with
TFo 10 seconds
andTH -0.52 Tm
where Tm is the time duration of a modulation period, an in-
crease in dwell time proportional to an increase in TFo by
a factor of cI results in an increase in SNR per slant by
the ratio, c2, as given by
(c1-TH/Tm) T mC. - (1_TH/T,.)Tm
Consequently, Eqn. (1.1-14) is modified to read
PA A v TFOS) -c4wR4 kT o NF o Lfl
As before, let Rz denote the range at which (S/N) becomes
unity. Recalling that TFo - do/VRo, it follows that
R rkT0 PAAa
R -, C2 4 kTo, NF o L fn V. A ° - c2 R1 AO
- R3 (cITH/Tm) (5.2-1)I(I-THTm)
228
Substitution of TH/Tm = 0.52 in the above equation yields
R4 M (1-0.52/c) c(5-R,=R-~-----~-k~'J(5.2-2)
Since A = CIAO, Eqn. (5.1-4) can be written as
R R 3CIAO - R 4'-"IAO) (5.1-4)
Note that Eqn. (5.2-2) differs from Eqn. (5.1-4) only by the
factor of
C2 (1-0.52/c,
Hence, the range at which SNR becomes unity is extended by
the factor of c2/cI.
To correct for the cell false alarm probability change as
a result of an increased dwell time, note that, once the
cell false alarm probability corresponding to the system
false alarm rate specification has been determined, the cell
false alarm probability is affected only by the time utili-
zation factor K1 according to Eqn. (2.5-25). The effect of
this change on Rz depends on the particular post detection
integration chosen.
The detection probability after post detection integra-
229
tion for Swerling case 1 and 3 targets is given in Section
3.2 as
Pd " joP;(M,N) p(S/N)d(S/N) • (3.2-7)
This is in the same form as Eqn. (3.2-12), the single slant
detection probability before post detection integration for
Swerling case 2 and 4 targets. For Swerling case 2, Pdi is
given by
Inpp. -exp ( ) . (3.2-13)
14{S7N)i
Therefore, for a Swerling case 1 target we try for the
purpose of approximating Eqn. (3.2-7) an equation of the
form
Pd exp [f](5.2-3a)where Pfa' the cell false alarm probability after 3 of 3
post detection integration, is given by
Pfa - Pf3
and (90/N) is the signal-to-noise ratio per CPI if all the
pulses are received through the antenna beam at its peak
gain. Taking the natural logarithm of Eqn. (5.2-3a) and re-
230
arranging yields
i (5.2-3b)
The value of c4 can be obtained from Eqn. (5.2-3b) and a
point from a Pd versus (S0/N) plot for a given value of pf.
Referring to Figure 3.3-3, the following values for two
points, each for two different values of pf, are read off
from the plots of Pd versus (90/N) for Swerling case 1 after
3 of 3 post detection integration
(S 0 /N) @ Pd-0. 3 (S0/N) @ Pd=0.5
* Pf-7.6xl0 - 4 15.85 (12.00 dB) 28.31 (14.52 dB)
@ Pf-7.6xl0 - 5 20.61 (13.14 dB) 36.64 (15.64 dB)
For pf-7.6xlO- 4 substitution of (S0/N)=15.85 and Pd= 0 .3
into Eqn. (5.2-3b) gives c4=1.062. Inserting this value of
c4 and (g0 /N)-28.31 into Eqn. (5.2-3a) yields Pd=0 .5. This
agrees with the plot of Figure 3.3-3 which is obtained by
numerical integration. Trying still other values of (S0/N)
-4for pf-7.6xl0 reveals that Eqn. (5.2-3a) is an excellent
approximation for Pd as a function of (S0/N) and pf. How-
ever, a different value of c4 is required for each value of
Pf. The value of c4 corresponding to pf=7.6xl0-5 is 1.0935.
231
Since what is sought is an approximate equation for in-
terpolation between two values of (S0/N) at a given value of
Pd due to a change in pf, an expression that is slightly
different from Eqn. (5.2-3b) which is less sensitive to
changes in pf is tried. The equation is of the form:
Pd =C . xp1 In _3ra I (5.2-3c)1*(So/N)
Associating points from the plots of Figure 3.3-3 with the
corresponding terms in Eqn. (5.2-3c), the value of c5 cor-
responding to pf=7.6xl0-4 and pf=7.6x10 -5 at a Pd level of
0.3 becomes 1.078 and 1.119, respectively. Using the aver-
age of these two values for c5 in Eqn. (5.2-3c) is found to
give accurate results relating changes in pf to the corre-
sponding changes in (S0/N) for a fixed value of Pd over a
range of change in pf by a factor of 10. This relation-
ship can be expressed as
Pd =C. expI- In]f1 mc.exp~ I-I Pf
1 + (pI/N), I + (So/N)2
Dividing both sides of the above equation by c5 and taking a
natural logarithm gives
F 1n(pfi [ 3 1]n(p f 2 ) 3
L 'SO/N) j L I(So,/N),
or
232
(Pf) 3 In Pf&I 1+(So/N) 1 (S3/N)I3, -for (S0/N),, (SO/N)z>1 . (5.2-3)
_ ( pf 2)s in Pf'' = 1+(So/N) 2 (S/N) 2
Note from Eqn. (2.5-25) that the ratio of false alarm prob-
abilities after 3 of 3 post detection integration equals the
inverse of the ratio of time utilization factors. Since the
optimization is to determine how much to slow down the an-
tenna scan rate with respect to the scan rate in the
baseline, pf must be reduced to maintain the same system
false alarm rate as the scan rate is reduced and, therefore,
the time utilization factor is increased.
In the vicinity of (S0/N)=12.2 dB and pf=7.6xl0- , ade
crease in pf by a factor of 10 results in a corresponding
decrease in the detection probability. This is equivalent to
maintaining pf at 7.6xi0-4 and reducing SNR by 1.2 dB, a
factor of approximately 1.32 as can be seen in Figure 3.3-3.
Another way of putting it is that an increase in SNR by a
factor of 1.32 is required to maintain the Pd level the same
as before. This can be verified by substituting these pf
values into Eqn. (5.2-3).
As mentioned previously, the ratio of false alarm prob-
abilities after 3 of 3 post detection integration equals the
233
inverse of the ratio of time utilization factors. Note that
c 2 is equal to the ratio of time utilization factors. Sub-
stitution of (1/c 2 ) for (Pfa /P fa) in Eqn. (5.2-3) yields
In P fl ~_ °IN ) , A C 3 a (5.2-4)
In -cl Pfa1) - N)z
For values of c1 equal to 1.1 and 10, the values of c2 are
1.098 and 1.975, respectively. The corresponding values of
c3 are 0.996 and 0.969. These values are the ratios by which
SNR is effectively reduced. Or equivalently, SNR must be
increased by the inverse of these values to maintain the
original level of Pd" Thus, it is seen that once the cell
false alarm probability is determined for the particular
post detection integration according to the procedure estab-
lished in Section 2.4, the false alarm probability change as
a consequence of an increase in dwell time has a small ef-
fect on the SNR. Hence, as a first approximation, it can be
ignored in the scan rate optimization using 3 of 3 post de-
tection integration. The effect of time overhead on other
post detection integration can be determined in the same
way.
When the dwell time is increased by a large factor, the
time overhead becomes a negligible portion of the modulation
period. Returning to Eqn. (5.2-2), note that
234
lr(1-0.52/cl)=.8cj-o 0.48 i 2. 08.
Its impact on R is (2.08)1/4 = 1.2 in the limit. This rep-
resents a 20 % increase in the detection range. However, the
maximization peaks are very broad. As a result, this factor
can be dropped for simplicity from Eqn. (5.2-1). Even so,
the effect of time overhead in the multi-slant processing is
included in evaluation of the detection and false alarm
probabilities for a specified post detection integration.
With respect to Eqn. (5.1-10) for Pc' all range units are
normalized by R1 which includes all fixed radar and target
parameters. Note that Pd(SO/N) can be converted to Pd (R/Rz)
by using the relationship
(R )4 (O/N),(SO/N)
Using Eqn. (5.1-4) R/Rz can be converted to R/R1 once A/RI
is specified:
andR, 4 = II
(So/N),
Hence, when R= Ri,
235
The above conversion allows the computation of Pd as a
function of the normalized range, R/RI . A set of values for
Pd(R/Rl) is computed according to the procedure outlined in
Chapter 3 for the normalized range increment, A/R1 , ranging
from 0.01 to 0.1. Pc(R/R1 ) is then computed for each value
of the normalized range increment chosen. From a set of
these values, the optimum A/RI and the corresponding Rm/Ri
are selected for Pc 0.9.
These optimum values for P -0.9 are listed in Table 5.2-1C
for Swerling case 1 through 4 and Marcum target models.
Table 5.2-1
Optimum Values A/R1 and the Corresponding Values of
Rm/R1 - 3 of 3 Post Detection Integration
PC = 0.9; P = 4.39x10 10
Swerling case Marcum1 2 3 4 nonfluctuating
A /R1 0.050 0.058 0.060 0.080 0.078
Rm/R1 0.157 0.132 0.171 0.171 0.223
236
The optimum values derived by Mallett and Brennan are
listed in Table 5.2-2 for Pc=0.9 for easy comparison.
Table 5.2-2
Optimum Values A/R and the Corresponding Values ofRm/R1 iy Mallett and Brennan
ml6PC = 0.9; P = ix10 6
Swerling case Marcum1 2 3 4 nonfluctuating
A/R 1 0.042 --- 0.053 --- 0.080
RmR 1 0.155 --- 0.169 --- 0.196
Note that the referenced values assume a single slant system
for which computation of Pd is fairly straightforward. It
should be observed that in a single slant system there is no
difference between a Swerling case 1 and a Swerling case 2
(nor a Swerling case 3 and a Swerling case 4). Hence, their
result does not show values for Swerling cases 2 and 4.
Still, the optimum values determined here for a system with
a particular post detection integration are found to be very
close to the corresponding values found by Mallett and
Brennan even though they are evaluated at different false
alarm probabilities. Therefore, performance improvement
based on cumulative detection probability at a level of 0.9
237
for a radially approaching target with a speed of Mach 1 can
be 7-8 dB as shown by an example in Section 5.1 when the an-
tenna scan rate is reduced by a factor of approximately 5.
In this example, note that the one minute time constraint is
not observed. Instead, the outer boundary of the cumulative
detection zone is taken as 400 nautical miles.
238
5.3 Detection Probability with Noncoherent Integration in
Lieu of Binary Integration
The Bayes likelihood ratio test for detecting targets
based on noncoherent integration of multiple observations in
white Gaussian noise is discussed in Radar Detection by
DiFranco and Rubin (2]. Specific forms of integral expres-
sions for the probabilities of detection resulting from bi-
nomial post detection integration were derived and presented
in Chapter 3. The integral expressions describing the prob-
ability of false alarm and probabilities of detection under
noncoherent integration are summarized below following the
work by Difranco and Rubin.
The optimum receiver structure in the region of small
signal-to-noise ratio is the matched filter for each pulse
followed by a square law detector and a noncoherent summer
or a video integrator for all four Swerling cases and Marcum
target models. Let Y denote the sufficient statistic. The
sufficient statistic, Y, is one half the sum of the square
of the envelop voltage ri,
NY I (1/2)r
where N is the number of observation samples noncoherently
integrated. For the kth doppler cell, note that the DFT co-
efficient X(k) which arises from an Ni-point FFT constitutes
a single sample for the noncoherent integration.
239
In the absence of the signal, the probability density of
Y under hypothesis 0 (target absent) assumes the form,
yN-1 e-Y / (N-I)!, Y>OP(YIO) = (5.3-2)0, Y<0.
This is a chi-square density with 2N degrees of freedom.
It arises because Y is the sum of 2N statistically indepen-
dent squared Gaussian random variables. That is, it is the
sum of N pairs of squared in-phase and quadrature Gaussian
noise components each having zero mean and unit variance
(i.e., normalized). Thus, the probability of false alarm is
given by
Pfa" 1 _Y -1 e-y/(N-1)!dY - I - I(Yb/I4 , N-i). (5.3-3)
where Yb is the threshold and I(u,s) is the incomplete gamma
function defined by
I(u,s) = (e-v vS/s!)dv.
The probability of detection is given by
Pd 4P(Yl)dY (5.3-4)
where p(Y1l) is the probability density of Y for Y>O under
hypothesis 1 (target present) with peak SNR per pulse, %p.
The relationship of the peak SNR to the envelope amplitude
A, and the probability densities of A and 5p for Marcum and
Swerling target models were given in Section 2.6.
240
The probability density of Y, p(YI1), and the probability
of detection for Marcum (nonfluctuating) and Swerling target
models are:
Marcum (nonfluctuating)
for Y>O
p(YIl) = (2 Y/NTP)(N-1)/2 e- Y - N /2 IN_1 ( 2 NV ), (5.3-5)
Pd= 1 - (2Y/ P) (N-1)/2 e-Y-N 5/2 IN_1("I N'Y)dY
= I - T A (2N-1,N-,j1 N P/2). (5.3-6)
IN_(x) is the modified Bessel function of the first kind,
order N-1 and TB(m,n,r) is the incomplete Toronto function,
TB(m,n,r) = 2rn-m+l e- tm -n e-t In (2rt)dt.
Swerling case 1
for Y>O
(1 + l/Nt2)N- 2 e-Y/(I+N F/2)p(Y1 ) -
YI[ I ) N-2]0, (5.3-7)
241
Pd-1 - I(YW4ITNi,N-2) + (l+lN;,/2)N-1 e-Y) /(1+Nlv/2)
.I[Ybl(TN-1(1+11NV%/2)),N-2). (5.3-8)
Swerlins case 2
for Y>O
p (y11) 1/ ll 4 zN (N-1)!JYNl eY/ (1+ tf2) (5.3-.9)
p (x e X)/(N.1)!dx
I I[Yb1(ofN(+tP12)),N-1]. (5.3-10)
Skwerling. cae 3
for Y2 O
1+1N4) 2 -i e( ' (1N / 4/4
-[(Y/ (1+1/Ni,/4) ArN-Fi) , N-2)+ (-!(1N/42
- W -1
(5.3-11)
242
Pd Pf p(Y) dY.
Swrling 4
for Y_>O
yN-1 e-Y/(1+ 4 ,4) N! ,
p(YI1) = 2
y k•k (5.3-12)
k! (N+k-1) ! (N-k)!,
I[ (Yb / ( 1-(R4 )i+) , N+k-i ]
=1 - N!/(l+-p7/4)N k(/4)k
Pd (IP/4)k:(N-k) :(5.3-13)
Numerical integration of above equations is used in Sec-
tion 5.4 to determine the threshold and probabilities of de-
tection under noncoherent integration.
243
5.4 Probabilities of Detection with Slower Scan Rate
In this section, M of N binary post detection integration
is compared with noncoherent integration (NCI) along with a
2 to 1 increase in dwell time as a means for detection en-
hancement. The 2 to 1 increase in dwell time is achieved by
reducing the antenna scan rate by a factor of 2. Enhancement
options examined include slant-to-slant frequency agility
and back-to-back antennas. When used, the back-to-back an-
tennas are assumed to be switched to the single receiver/
transmitter one at a time every 180 degree rotation of thR
rotodome so that the azimuth coverage is reduced from 360 to
180 degrees and the same 180 degree azimuth sector is cov-
ered with twice the dwell time. It was shown in Section 5.1
and 5.2 that scan rate optimization called for a consider-
ably slower scan rate. Reduction of the scan rate by more
than a factor of 2, however, is prevented by track update
considerations for near-in targets. For post detection inte-
gration, it is well known that noncoherent integration is
more efficient than binary integration. Only recent advances
in signal processing hardware make it feasible to implement
NCI in a high PRF radar.
The single scan detection probabilities for 3 of 3, and 3
of 6 post detection binary integration, and noncoherent in-
244
tegration with 3 and 6 observation samples, all within the
same false alarm time constraints but with the antenna scan
rate at 3 rpm, are first determined for the Swerling case
1, 2, and Marcum target models. The false alarm constraint
is such that the system false alarm rate is 2 or less in a
10 second interval on the average. It should be noted that
Swerling case 2 results apply to Swerling case 1 when
slant-to-slant frequency agility is used in the radar op-
eration. Under this condition, Swerling case 1 is trans-
formed to Swerling case 2. The single scan detection prob-
abilities for the four processing options are plotted for
the Swerling case 1, 2 and Marcum target models in Figures
5.4-1 through 5.4-4. The numerical results with the same
processing options for Swerling case 1 and 2 along with the
baseline performance are also shown in Table 5.4-1. For gen-
erating these results, an average beam shape loss is used
for all cases, first, to facilitate the detection probabil-
ity evaluation with NCI where equal signal-to-noise ratios
are assumed for all samples, and second, to generate results
on the same basis for all processing options considered.
The same reference target located at the reference range,
as was used previously, is employed in this chapter for es-
tablishing the reference SNR for the particular processing
configuration. For the baseline 3-slant configuration with a
245
CNJ cv
E-4'
oo $
1 1
ot
41 010 .4
CCrn,
Lin
*co00 00 0 0 0 0 00 0
uoiloala jo Cllq-qAc
246
ANN
CCID
oo 0---- ---- --- ---- ---- -I1
1.4
4)
__ A1to r-
ci0
-% - --HNS
zU)
o o o~4
uoilaal jo Sllqeoc
247
C14
CC
00
0C.Z 4J
-r
0 E-
os 00-
C) >
cP4Z
0 ~ ~ ~ ~ ~ ' -z 0 0000
o M M r O 0 t 4 ,- n' N\ '- 0 -S )
~.6 C C C C C C C C C
uorpajaj jo SllleqAc
248
CNN
o 04
00
- - - - - - - co n
E-4GIn -- - - -- _ _ 0 C) 4.I
r7 * a~
0
0 4
0. IS
'-000000 00uo~~V-q zo
C24
?able 5.4-1 Single Scan Detection Probabilities 13 rpa iatena Scan late vith lorsal live Overhead
4444..4leference SRI 17.IdB# ........................... Reference SKI : 12.2 dB ------------.....-.. 3 ICI ----- sus 3 Of 3sts 6 IC -------- 3 Of 6 ----- 3 of 3 / 6 rpm
(baseline)
(So/1) Si-i S1-2 SI-i SI-2 SI-I Si-2 SI-i SI-2 SI-i SI-2
3.00 0.128 0.074 0.090 0.011 0.337 0.376 0.263 0.189 0.094 0.0128.20 0.139 0.086 0.093 0.013 0.354 0.11 0.284 0.212 0.103 0.0148.40 0.151 0.099 0.109 0.015 0.370 0.448 0.300 0.236 0.113 0.0166.60 0.164 0.113 0.119 0.018 0.386 0.485 0.316 0.261 0.123 0.0198.80 0.177 0.128 0.130 0.021 0.403 0.522 1.332 0.287 0.134 0.0221.00 0.190 0.145 0.141 0.024 0.419 0.558 0.348 0.315 0.146 0.0263.20 0.204 0.164 0.153 0.028 0.435 0.594 0.364 0.343 0.158 0.0299.40 0.218 0.183 0.165 0.032 0.451 0.628 0.380 0.373 0.170 0.0349.60 0.232 0.204 0.178 0.036 0.467 0.662 0.397 0.403 0.183 0.0389.80 0.247 0.226 0.191 0.041 0.483 0.594 0.413 0,433 0.196 0.043
10.00 0.262 0.249 0.204 0.046 0.499 0.724 0.429 0.463 0.210 0.04910.20 0.278 0.273 0.218 0.052 0.514 0.753 0.445 0.494 0.224 0.05510.40 0.293 0.298 0.232 0.059 0.529 0.780 0.461 0.524 0.238 0.06210.60 0.309 0.324 0.247 0.066 0.544 0.305 0.477 0.554 0.253 0.06910.80 0.325 0.351 0.262 0.073 0.559 0.828 0.492 0.584 0.268 0.07611.00 0.341 0.377 0.277 0.081 0.574 0.849 0.508 0.613 0.283 0.08511.20 0.358 0.405 0.292 0.090 0.588 0.868 0.523 0.641 0.299 0.09311.40 0.374 0.432 0.308 0.099 0.602 0.885 0.538 0.668 0.314 0.10311.60 0.390 0.460 0.324 0.109 0.616 0.900 0.553 0.694 0.330 0.11311.80 0.406 0.488 0.340 0.119 0.629 0.914 0.568 0.719 0.346 0.12312.00 0.423 0.515 0.355 0.130 0.842 0.926 0.582 0.743 0.362 0.13412.20 0.439 0.542 0.372 0.141 0.655 0.937 0.596 0.765 0.378 0.14612.40 0.455 0.569 0.388 0.153 0.667 0.946 0.610 0.786 0.394 0.15812.60 0.471 0.595 0.404 0.165 0.679 0.955 0.623 0.806 0.410 0.17012.80 0.487 0.621 0.420 0.1 8 0.691 0.962 0.636 0.824 0.426 0.18313.00 0.502 0.645 0.436 0.191 0.702 0.968 0.649 0.842 0.442 0.19713.20 0.517 0.669 0.452 0.205 0.714 0.973 0.662 0.858 0.458 0.21013.40 0.533 0.692 0.467 0.219 0.724 0.978 0.674 0.872 0.473 0.22513.60 0.548 0.714 0.483 0.234 0.735 0.951 0.686 0.886 0.489 0.23913.80 0.562 0.735 0.499 0.248 0.745 0.985 0.697 0.898 0.504 0.25414.00 0.577 0.755 0.514 0.263 0.755 0.987 0.709 0.909 0.520 0.26914.20 0.591 0.774 0.529 0.279 0.764 0.990 0.720 0.919 0.535 0.28514.40 0.605 0.792 0.544 0.294 0.774 0.991 3.730 0.929 0.549 0.30014.60 0.618 0.809 0.558 0.310 0.782 0.993 0.740 0.937 0.564 0.31614.80 0.632 0.825 0.573 0.326 0.791 0.994 0.750 0.945 0.578 0.33215.00 0.645 0.840 0.587 0.342 0.799 0.995 0.760 0.951 0.592 0.34815.20 0.657 0.854 0.601 0.358 0.807 0.996 0.769 0.9W7 0.606 0.36415.40 0.670 0.867 0.614 0.374 0.815 0.997 0.778 0.963 0.619 0.38015.60 0.682 0.879 0.628 0.390 0.823 0.998 0.787 0.967 0.633 0.39615.80 0.693 0,890 0.641 0.407 0.830 0.998 0.796 0.971 0.645 0.41316.00 0.705 0.900 0.653 0.423 0.837 0.998 0.804 0.975 0.658 0.42916.20 0.716 0.909 0.666 0.439 0.844 0.999 0.812 0.978 0.670 0.44516.40 0.726 0.918 0.678 0.455 0,850 0.999 0.819 0.981 0.682 0.46116.60 0.737 0.926 0.689 0.471 0.856 0.999 0.827 0.984 0.694 0.47616.80 0.747 0.933 0.701 0.486 0.862 0.999 0.834 0.986 0.705 0.49217.00 0.757 0.940 0.712 0.502 0.868 0.999 0.840 0.988 0.716 0,50717.10 0.760 0.943 0.717 0.509
250
Table 5.4-1
Single Scan Detection Probabilities 13 rpm lntena Scan late uith lortal Tine Overhead (cont'd)
+... ++Reference SIB 17.ldB++ ....4 .........------------l eference Sit : 12.2 dB ----------------------- 3 ICI .--- s-s33 3 01 3s:ssts 6 ICl ------- 3 0! 6 ----- 3 of 3 / 6 rpu
(baseline)(*/l) S-1 SI-2 Si-i SU-2 SI-1 S-2 S1-1 SI-2 Si-i SI-2
17.20 1.788 0.048 0.723 0.517 1.873 1.000 1.147 1.990 0.727 0.52317.40 0.775 0.952 0.733 0.532 1.879 1.000 1.853 1.991 0.137 0.53811.60 0.784 0.957 0.743 0.547 1.884 1.000 6.859 1.992 0.747 0.55317.80 0.793 0.961 0.753 0.562 1.189 1.000 0.865 0.993 0.757 0.56718.00 0.801 0.965 0.763 0.576 1.893 1.000 0.871 0.994 0.766 0.58118.20 0.809 0.969 0.772 0.590 1.898 1.000 0.876 0.995 0.775 0.59518.40 0.817 0.972 0.781 0.604 3.902 1.000 0.881 0.996 0.784 0.60918.60 0.824 0.975 0.789 0.618 0.906 1.000 0.886 0.996 0.793 0.62318.80 0.831 0.978 0.198 0.631 0.910 1.000 0.891 0.997 0.801 0.63619.00 0.838 0.980 0.806 0.644 0.914 1.000 0.896 0.997 0.809 0.64919.20 0.845 0.983 0.814 0.657 0.918 1.000 0.900 0.998 0.816 0.66119.40 0.851 0.984 0.821 0.669 6.922 1.000 0.905 0.998 0.824 0.67319.60 0.857 0.986 0.828 0.681 0.925 1.000 0.909 0.998 0.831 0.68519.80 0.863 0.988 0.835 0.693 0.928 1.000 0.913 0.999 0.838 0.69720.00 0.869 0.989 0.842 0.704 0.931 1.000 0.916 1.199 0.845 0.70820.20 0.874 0.990 0.849 0.715 0.934 1.000 0.920 0.999 0.851 0.71920.40 0.880 0.992 0.855 0.726 0.937 1.000 0.923 0.999 0.857 0.72920.60 0.885 0.992 0.861 0.736 6.940 1.000 0.927 0.999 0.863 0.74020.80 0.890 0.993 0.867 0.746 1.942 1.000 0.930 0.999 0.869 0.75021.00 0.894 0.994 0.872 0.756 0.945 1.000 0.933 1.000 0.874 0.75921.20 0.899 0.995 0.877 0.765 0.947 1.000 0.936 1.000 0.879 0.76921.40 0.903 0.995 0.883 0.775 1.950 1.010 0.939 1.000 0.884 0.77821.60 0.907 0.996 0.888 0.783 5.952 1.000 0.941 1.000 0.889 0.78721.80 0.911 0.996 0.892 0.792 0.954 1.000 0.944 1.00 0.894 0.79522.00 0.915 0.997 0.897 0.800 6.956 1.000 0.946 1.000 0.899 0.80322.20 0.919 0.997 0.901 0.808 6.958 1.000 0.949 1.000 0.903 0.81122.10 0.922 0.998 0.905 0.816 6.960 1.000 0.951 1.000 0.907 0.81922.60 0.925 0.998 0.910 0.823 1.962 1.000 0.953 1.000 0.911 0.82622.80 0.929 0.998 0.913 0.831 0.963 1.000 0.955 1.000 0.915 0.83323.00 0.932 0.998 0.917 0.837 0.965 1.000 0.957 1.000 0.91t 0.84023.20 0.935 0.999 0.921 0.844 5.966 1.000 0.959 1.000 0.922 0.84623.40 0.938 0.999 0.924 0.851 0.968 1.000 0.961 1.000 0.925 0.85323.60 0.940 0.999 0.927 0.857 1.969 1.000 0.962 1.000 0.929 0.85923.80 0.943 0.999 0.930 0.863 1.971 1.000 0.964 1.000 0.932 0.86524.00 0.945 0.999 0.933 0.868 1.972 1.000 0.966 1.000 0.935 0.87024.20 0.948 0.999 0.936 0.874 0.973 1.000 0.967 1.000 0.937 0.87624.40 0.950 0.999 0.939 0.879 1.974 1.000 0.969 1.000 0.940 0.88124.60 0.952 0.999 0.942 0.884 1.976 1.000 0.970 1.000 0.943 0.88624.80 0.954 0.999 0.944 0.889 1.977 1.000 0.971 1.000 0.945 0.89125.00 1.956 1.000 0.947 0.894 1.978 1.000 0.973 1.000 0.948 0.89525.20 0.958 1.000 0.949 0.898 1.979 1.000 0.974 1.000 0.950 0.20025.40 0.960 1.000 0.951 0.903 1.110 1.000 0.975 1.000 0.952 0.90425.60 0.962 1.000 0.953 0.907 1.981 1.000 0.976 1.000 0.954 0.90825.80 0.964 1.000 0.955 0.911 0.981 1.000 0.977 1.000 0.956 0.91226.00 0.965 1.000 0.957 0.915 0.982 1.000 0.978 1.000 0.958 0.916
251
scan rate of 6 rpm, this results in a single slant reference
SNR, (S0/N)0 , of 12.2 dB. Note that the reference SNR for
a 6-slant configuration with a scan rate of 3 rpm is also
12.2 dB. For a 3-slant configuration with a scan rate of 3
rpm, (S0/N)0 equals 17.1 dB. The increase in (S0/N)0 when
the scan rate is reduced by a factor of 2 is more than 3 dB
because the number of pulses integrated increases without an
increase in the time overhead.
The analysis result shows that a considerable improvement
is possible in the single scan detection probability with
NCI and slant-to-slant frequency agility when they are ac-
companied with an increased dwell time. Frequency agility
renders a rapid fluctuation to Swerling case 1 (and case 3)
targets such that they behave as Swerling case 2 (or case 4)
targets. Single scan detection probability with NCI im-
proves rapidly for Swerling case 2 targets in the high SNR
region. Increased dwell time, as a consequence of slower
scan rate, places the returned signal from the reference
target at the reference range in the higher SNR region.
Since a different processing configuration gives rise to a
different level of SNR per slant for a given target situa-
tion, the reference SNR for each configuration is used to
identify one point in the Pd versus (S0/N) curve to a
I252
particular detection range, namely, the reference range of
the reference target. Then, the remaining values of (S0/N)
can be converted to the corresponding range using the radar
range equation.
Improvements in single scan detection probabilities with
an increase in dwell time are, of course, expected since SNR
is proportional to the dwell time. Whether the improvement
in single scan detection probability with slower revisit
rates leads to improvement in cumulative detection probabil-
ity and track update rates is the real question. Since a
slower scan rate results in fewer opportunities for the cu-
mulative detection process and track updates, a back-to-back
antenna configuration is also included as an option which is
used to cover a reduced surveillance sector of 180 degree
azimuth. In the following investigation, the target is as-
sumed to be an aircraft whose RCS fluctuates according to
the Swerling case 1 model. Results for other target models
can be determined in the same manner.
The improvement in cumulative detection performance is
determined on the basis of achieving Pc = 0.9 in 60, 20, and
10 seconds using the single scan detection probabilities
shown in Table 5.4-1. The track update performance is deter-
253
mined on the basis of the average number of 'hits' in a 5
minute time interval. It will be shown that the slower scan
rate improves performance in both cumulative detection prob-
ability and track update rate for the outer ranges at the
expense of a poorer track update rate in the inner ranges
when compared to the result with normal scan rate used in
the baseline. This is because update rate can never exceed
the antenna scan rate.
Assuming that target range closure is negligible during
the interval over which a specified value of Pc is achieved,
PC is given by
PC l-(l-Pd ) L
where L is the number of revisits to or scans by the target
in question during the cumulative detection interval. It
follows that the necessary level of Pd to achieve a
specified value for Pc can be expressed as
Pd = l-exp[(l/L)ln(l-Pc)] . (5.4-1)
The required Pd that yields Pc = 0.9 is calculated from
Eqn. (5.4-1) as a function of L and is listed in Table
5.4-2.
254
Table 5.4-2
Required Pd to achieve Pc = 0.9 in L scans
L P
1 0.900
2 0.683
3 0.535
4 0.437
5 0.369
6 0..318
The improvement in dB is the differential margin in SNR
relative to the baseline for the reference target located at
the reference range for each processing option. By defini-
tion, the SNR margin is the amount by which the reference
SNR is higher than the SNR required to yield a level of Pd
that will translate to the specified Pc. This method of per-
formance comparison was also used in Section 4.5. The
concept is illustrated in Figure 4.5-4.
At an antenna scan rate of 3 rpm, the number of scans in
20 and 60 seconds are 1 and 3, respectively. These values of
255
L require Pd levels of 0.9 and 0.535, respectively, to yield
PC = 0.9 in 20 and 60 seconds. As an illustration of the
margin computation, it is seen from Table 5.4-1 under 3 NCI
for Swerling case 2 (which would apply to Swerling case 1
when slant-to-slant frequency agility is used) that the re-
quired (S01 N) for Pd = 0.9 and Pd - 0.535 is 16.0 dB and
12.1 dB. These translate to SNR margins relative to the ref-
erence SNR of 17.1-16.0=1.1 dB and 17.1-12.1=5.0 dB. In the
baseline with 3 of 3 post detection integration and with the
antenna scan rate of 6 rpm, the required Pd for PC = 0.9 in
20 and 60 seconds is 0.683 and 0.318, respectively, with the
corresponding (S0/N) requirement of 16.4 dB and 11.4 dB. The
SNR margin in each case is 12.2-16.4=-4.2 dB and
12.2-11.4-0.8 dB. By comparing these margins to the margins
under 3 NCI with frequency agility at an antenna scan rate
of 3 rpm, the improvement is seen to be 5.3 dB and 4.2 dB,
respectively, for PC = 0.9 in 20 and 60 seconds.
Performance improvements obtained in this manner for sev-
eral signal processing options considered for a Swerling
case 1 target are summarized in Table 5.4-3. The options in-
clude with and without frequency agility and with and with-
out back-to-back antennas. Note that the degree of improve-
ment depends on the performance criterion adopted.
256
Table 5.4-3
Summary of Performance Improvement with the AntennaScan Rate of 3 rpm for a Swerling Case 1 Target
a. antenna scan rate of 3 rpm (360 coverage in azimuth)
P in 60 sec. P in 20 sec.req'd P 0.535 0.90d SNR(dB) margin gain* SNR(dB) margin gain*
(w/o frequency agility)3 NCI 13.4 3.7 2.9 21.2 -4.1 0.13 of 3 14.3 2.8 2.0 22.1 -5.0 -0.83 of 6 11.4 0.8 0.0 19.2 -7.0 -2.86 NCI 10.5 1.7 0.9 18.4 -6.2 -2.0
(w/ frequency agility) -3 NCI 12.1 5.0 4.2 16.0 1.1 5.33 of 3 17.4 -0.3 -1.1 25.4 -8.3 -4.13 of 6 10.5 1.7 0.9 13.8 -1.6 2.66 NCI 8.9 3.3 2.5 11.6 0.6 4.8
b. anteana scan rate of 3 rpm with a back-to-back antenna(180 coverage in azimuth)
PC in 60 sec. Pc in 20 sec.req'd Pd 0.535 0.683
SNR(dB) margin gain* SNR(dB) margin gain*(w/o frequency agilility)
3 NCI 10.8 6.3 5.5 15.5 1.6 5.83 of 3 11.5 5.6 4.8 16.4 0.7 4.93 of 6 8.6 3.6 2.8 13.5 -1.5 2.76 NCI 8.0 4.2 3.4 12.7 -0.5 3.7
(w/ frequency agilility)3 NCI 10.6 6.5 5.7 13.3 3.8 8.03 of 3 14.7 2.4 1.6 19.6 -2.5 1.73 of 6 9.0 3.2 2.4 11.5 0.7 4.96 NCI 7.8 4.4 3.6 9.7 2.5 6.7
c. antenna scan rate at 6 rpm
PC in 60 sec. Pc in 20 sec.req'd Pd 0.318 0.683
SNR(dB) margin gain SNR(dB) margin gain
3 of 3 11.4 0.8 ref 16.4 -4.2 ref
*compared to the corresponding SNR margin with 3 of 3 pro-cessing @ antenna scan rate of 6 rpm
257
If the criterion is the track update rate instead, which
is defined as the blip scan ratio multiplied by the number
of scans in a specified time interval, improvement figures
different from those shown in.Table 5.4-3 result. For ex-
ample, at the range-where Pc = 0.9 is reached, the average
number of track updates (hits) in 5 minutes for Pc intervals
of 20 and 60 seconds at the antenna scan rate of 3 rpm are
13.5 and 8, respectively. The corresponding average numbers
of track updates at the antenna scan rate of 6 rpm are 20.5
and 9.5.
To yield the same level of track update rate at 6 rpm, a
Pd level of only 0.45 and 0.27 is required for Pc intervals
of 20 and 60 seconds, respectively. Corresponding values of
the required (S0/N) are 13.1 dB and 10.8 dB. Again, by com-
paring SNR margins, the improvement figure is determined to
be 2.0 dB and 3.6 dB for Pc intervals of 20 and 60 seconds,
respectively. Even though the improvement figures are
smaller, it is significant that the track update rate is im-
proved at this range by slowing down the scan rate. This im-
provement increases at ranges further out and decreases at
ranges closer in. This subject is further discussed at the
end of the Chapter.
258
Now assume that the azimuth coverage is reduced from 360
to 180 degrees. With a back-to-back antenna configuration
where the receiver/transmitter set is switched to one or the
other antenna every 180 degrees of the rotodome rotation sr
that the same 180 degree azimuth sector is covered by both
antennas, the revisit rate at 3 rpm for the covered sector
is the same as that provided by a single antenna at a scan
rate of 6 rpm. Therefore, the performance improvement mea-
sure based on Pc is the same as that based on the average
number of hits. Following the same procedures as above, the
equivalent dB improvement figures with frequency agility and
3 NCI with the back-to-back antenna are 8.0 dB and 5.7 dB
based on 0.9 Pc in 20 and 60 seconds, respectively.
These results demonstrate the efficiency of NCI in the
region of high SNR or high probability of detection rendered
by the slower scan rate when used in conjunction with fre-
quency agility. The improvement would be even greater had
the highly desirable track update rate of once every 10 sec-
onds been required. This would require a single scan Pd of
0.9. With two back-to-back antennas rotating at 3 rpm and
slant-to-slant frequency agility, this level of performance
is achievable at a range equal to 1.06(R0 ) for a 3 NCI and
at 0.96(R 0) for a 6-NCI where (R0 ) denotes the reference
259
range. This results from the the direct conversion of
one-fourth power of the SNR margin available at (R0 ), which
are 1.1 dB and -0.6 dB, respectively, for 3 NCI and 6 NCI.
In che baseline 3 of 3 post detection integration, Pd = 0.9
is rarely reached due to range eclipsing even when a target
is at a close range. Ignoring range eclipsing for ease of
comparison of other effects on detection performance, the
SNR margin for 0.9 Pd at the reference range is -9.9 dB.
Thus, the equivalent dB improvement is 11.0 dB and 9.3 dB
for 3 NCI and 6 NCI, respectively.
To facilitate the performance comparison, Pc values for
20 and 60 seconds are plotted in Figures 5.4-5 anC 5.4-6,
respectively, as a function of the mean normalized SNR, x,
defined as (S0/N) divided by the respective (S0/N)0 . This
eliminates the need for computing SNR margins. Comparison
on the basis of track update rate can be generated
similarly. Let u denote the average number of hits in a
specified time interval, say 5 minutes. Since there are 15
and 3t antenna scans in a 5 minute interval for 3 and 6 rpm
antenna scan rates, respectively, u as a function of the
normalized SNR, x, is qiven by
u(x) =(15)Pd(X)(So/N) o]: antenna scan rate at 3 rpm
(30) [Pd(x) (So/N)o]: antenna scan Late at 6 rpm.
260
N --- --- ( CC.
OR~op
* 0* C ,4
cN
0
0 0 0 0 C0 0 00 0 0 00~ 0) 0 0 (0 0)~ ) CJ '
SjTjq~qoij~ uoil3alap ;DAqIfITUCITIO
261
0
Neu
14 U
a. caI.,.
S '.4
z ---- U,
CF
-0 0~~ LC) 0 0 0 00 ~ ~ ~ - W O- 4 "
-4 -4 N i 0* 6
Sl~llqvqo Sz 4 uola.iaielun
0) 2620
The plots of the average number of track updates in a 5
minute interval with 3-NCI at 3 rpm for Swerling case 1 and
2 targets together with the same in the baseline 3 of 3 pro-
cessing at 6 rpm for Swerling case 1 targets are shown in
Figure 5.4-7. These figures clearly demonstrate that not
only is the Pc range extended with a slower scan rate but
the track update rate is also improved in the outer range.
263
$4 0 ft10~
to 4"'
EEn.
0
E0
V 00
N CN ( ( - V- N
*zu 9 asalpf Yifjo'NrReIA
-26
CHAPTER 6
SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK
6.1 Summary
This report presents results of an investigation
into the effectiveness of (a) scan-to-scan processing and
(b) scan rate reduction in improving the detection perfor-
mance of an existing airborne surveillance radar subjected
to power-aperture product and system false alarm con-
straints. As background, considerations involved in a per-
formance comparison of the baseline and modified radar con-
figurations were clearly explained where the primary
performance criterion was the cumulative detection probabil-
ity. Following an introduction in Chapter 1 of the radar
system under investigation together with basic concepts as-
sociated with the detection problem in a long range airborne
surveillance radar, those parameters which go through
changes with the system modification were identified and
methods for quantifying those parameters were developed in
Chapter 2. These included the number of coherent processing
intervals in a beam dwell given the search frame time, the
number of available pulses in a coherent processing inter-
val, beam shape loss, false alarm probability allocation,
and target models. It was shown how two different approaches
to false alarm calculations, one proposed by Marcum and the
other by Barton and Skolnik, could be used to relate cell
265
false alarm to system false alarm in a complex high PRF ra-
dar.
The baseline radar configuration was then analyzed in
Chapter 3. Probability density functions for the sufficient
statistic appearing in the likelihood ratio test were devel-
oped. Expressions for the detection probabilities after M of
N post detection binary integration were derived. These re-
sults were then averaged with respect to the assumed target
radar cross section fluctuations to obtain the expected de-
tection probabilities for five different target models.
Graphs of detection probability versus signal-to-noise ratio
were generated and presented for the five target models.
The scan-to-scan processing (SSP) was dealt with on a
theoretical basis (as opposed to simulation) in Chapter 4.
The baseline radar configuration was modified by incorporat-
ing two different versions of scan-to-scan processing: One
was a conventional J of K SSP while the other was a modified
J of K SSP. The philosophy is to lower the threshold to en-
able detection of extremely weak targets or extend the de-
tection range for conventional targets. This results in a
drastic increase of false alarms. These are suppressed by
requiring J detections in a K scan wide correlation window
associated with L total scans. The size of the correlation
window was determined by imposing limits consistent with
realistic target maneuvers. An analysis was carried out to
266
relate cell false alarm probability to system false alarm
rate when scan-to-scan processing is incorporated. The cor-
responding cumulative detection probabilities are derived.
It is shown, contrary to widely held optimistic projections,
that only a very marginal improvement results over the cumu-
lative detection probability of the baseline radar.
With a power-aperture product constraint, the only ad-
justable system parameter available for improving perfor-
mance is the scan rate over the surveillance volume. In
Chapter 5 previous work on scan rate optimization for a low
PRF radar is reviewed. The results suggested that improve-
ment could be achieved by slowing down the scan rate for a
given surveillance volume. An example was worked out in the
investigation to show that slowing down the scan rate in the
baseline radar could result in significant improvement. As
a result, the previous scan rate optimization for the low
PRF radar was extended to the high PRF case. Because of
track update requirements, scan rate reduction was limited
to a factor of two compared to what was specified for the
baseline radar. Both binary M of N and noncoherent post de-
tection integration were compared in conjunction with the
reduced scan rate in the analysis. Noncoherent post detec-
tion integration combined with a reduced scan rate was shown
to give rise to significant detection performance improve-
ment.
267
6.2 Recommendations for Future Work
The main body of this investigation dealt with detection
of Swerling and Marcum target models embedded in white
Gaussian noise. This is a problem encountered in a well de-
signed high PRF radar where the majority of targets of in-
terest fall in the clutter free doppler zone. The solution
to this problem is generally assumed in the literature to be
well known. However, the problems posed and the concepts and
methodology developed in this investigation while critical
in addressing the issues facing modern surveillance radar
problems today, cannot be readily found in the literature.
One of the major areas of radar research activity over
the past decade appears to center on optimum detection in
non-Gaussian interference, the principal source of the
non-Gaussian interference being clutter. Frequently, the in-
stantaneous power of radar returns from land clutter is
characterized as lognormal [10] or Weibull distributed [16),
while that from sea clutter is K-distributed [17]. This led
to a flurry of research activity into detection schemes
based on stochastic estimation, which is an extension of the
optimum detection theory [18], or Locally Optimum Detector
(LOD) [19) or Asymptotically Optimum Detector [20] on the
one hand, and adaptive clutter cancelling techniques on the
other [21, 22].
268
While these topics are interesting, detection approaches
under non-Gaussian clutter may be irrelevant to a majority
of the detection problems in clutter. It is indeed true
that the aggregate of returned clutter power observed over
the entire surveillance volume can assume lognormal,
Weibull, or K-distributed density. However, the conditional
density given the local mean of the clutter power taken from
one radar resolution cell over the time period during which
each detection decision is made within a radar scan is
likely to be either exponential or Rician. The assumption of
exponential or Rician density in turn depends on whether or
not the entire sequence of radar pulses is transmitted at
the same carrier frequency [23].
Describing the probability density of the entire set of
returned clutter power in a surveillance sector as an inte-
gral of the conditional probability density of the clutter
given its local mean multiplied by the probability density
of the mean using the mean as the variable of integration
(see Eqn. 6.2-1, p276) is based on the point of view arising
from the nonstationary characterization of clutter (24, 25,
26, 27]. Not only does this approach provide a means to ex-
plain a more complex form of clutter distribution, but more
importantly it allows the correlation property of the clut-
ter to be correctly modeled. Stating it differently, the
voltages associated with the in-phase and quadrature compo-
nents of the clutter returns during a beam dwell from a
269
given range-azimuth resolution cell is likely to be
Gaussian, albeit nonzero mean, when the radar resolution
cell intersects the ground patch which contains many elemen-
tal clutter cells. An elemental clutter cell is that clut-
ter patch bounded by its spatial correlation distance. Re-
turns from elemental cells of a ground patch may be also
Weibull (23] or Lognormal distributed [28]. However, when
the radar resolution cell encompasses many elemental cells,
the central limit theorem evidently is at work to render the
distribution of the received clutter power to be exponential
or Rician conditioned on its mean taking a certain value.
The assumption of multiple elemental cells will fail to hold
eventually as the radar resolution cell is made sufficiently
small as in a synthetic aperture radar (SAR) used for ground
imaging. Some authors report that the Gaussian assumption
also fails to hold at very low grazing angles.
Thus, for the majority of practical situations the detec-
tion problems in Weibull, lognormal, or K-distributed clut-
ter usually break down to problems of detection in Rayleigh
or Rician clutter (for the envelope voltage after combining
the in-phase and quadrature components) during each decision
interval (25]. The detection decision is made with an adap-
tive threshold that yields the desired constant false alarm
rate throughout the surveillance volume. The expected detec-
tion probability over the entire surveillance volume is ob-
tained by averaging the local detection results over the
270
variation of the clutter mean. Note that this process is
exactly the same as the procedure for obtaining the expected
detection probability after post detection integration of
Swerling case 1 and 3 targets (slowly varying with respect
to a CPI) in white Gaussian noise. This process was ex-
plained in Chapter 3. Totally irrelevant detection predic-
tions would result for Swerling case 1 and 3 target models
when a method appropriate for Swerling case 2 and 4 were ap-
plied to these models. Similarly, an optimum detector under
non-Gaussian interference produces an incorrect result when
the local interference over the detection decision interval
is Gaussian.
A preliminary analysis of clutter samples collected with
a high PRF radar is presented in Section 6.2.1. The result
tends to support the nonstationary characterization approach
to clutter. In particular, the mean and variance of the
clutter samples are found to vary significantly from one
range cell to the next.
As for the adaptive clutter canceller, which is an adap-
tive implementation of an optimum filter in the sense that
it maximizes the output signal-to-clutter-plus-noise ratio,
its predicted performance in comparison to that of a conven-
tional processor (MTI followed by a windowed FFT) is usu-
ally based on artificially simple assumptions which may be
irrelevant to the real situation encountered. Much of the
271
recent adaptive clutter canceller work which is of sig-
nificance is contained in MTI RADAR edited by Schleher [21]
and Optimized RADAR PROCESSORS by Farina [22]. Simplistic
assumptions typically made are:
1. The interference is a zero mean Gaussian random pro-
cess.
2. The interference has an exponential autocorrelation
function and the covariance matrix can be normalized
such that it is a positive definite hermitian matrix
with unity diagonal elements.
3. The number of pulses used for processing is small
(usually 8 or 16 pulses).
4. Adjacent range gate samples are independent and iden-
tically distributed so that the signal free sample
covariance matrix can be obtained by averaging a num-
ber of adjacent range gate samples.
While assumptions 1 and 2 are reasonable for an adaptive
array design in the presence of sidelobe clutter interference
[29), the mean of mainbeam clutter samples over a CPI is
hardly ever zero, and the sample correlation matrix does not
have identical diagonal elements (i.e., nonstationary). As-
sumption 3 is valid for a very simple low PRF radar. In a me-
dium or high PRF radar the number of pulses integrated is
much larger which, when coupled with the fact that the
272
clutter correlation matrix is almost singular (i.e., clutter
returns are highly correlated between pulse-to-pulse), makes
it difficult to invert or decompose the correlation matrix of
a large dimension. As for assumption 4, the clutter returns
in adjacent range cell samples taken with a high PRF radar
show that they are not identically distributed. In fact, the
mean and variance of the clutter samples exhibit significant
variations from one range cell to the next so that they can-
not be used to form a sample correlation matrix. This is
also reported to be true for a low PRF radar. the sample
correlation matrix can be formed from temporal samples in the
same range cell. The signal free. correlation matrix would be
difficult to obtain by this means, however.
A performance comparison of different clutter cancellers
is given in Secton 6.2.2. An infinite impulse response (IIR)
filter giving shaped velocity response, a finite impulse re-
sponse (FIR) filter using binomial coefficients as its
weights and and an FIR filter that derives its weights from
the eigen vector corresponding to the minimum eigen value of
the clutter correlation matrix are compared using actual high
PRF clutter data as inputs. No one approach was found to be
superior to the others tried.
In view of the lack of noticeable improvement in perfor-
mance of the optimum linear clutter canceller with real clut-
ter data, it seems reasonable to consider an option alternate
273
to the classical approach examined. This alternate approach
views the clutter as an a priori unknown, but deterministic,
process. Observation of a time sequence of clutter returns
reveals a slow monotonic variation of clutter amplitude over
a CPI. There is no random jump in amplitude or phase from
pulse to pulse. One possible explanation for this phenomenon
can be that the mainbeam clutter is dominated by a few
discrete scatterers whose signal amplitudes are slowly
modulated by the antenna scanning and platform motion. A
smooth curve fit to, or a low pass replica of, the clutter
amplitude and phase variation over a CPI can be made after
reception of the data and subtracted from the original data,
which in effect adaptively takes out a majority of the unde-
sirable clutter returns leaving uncancelled target signals
separated from the clutter doppler. Limited trials with col-
lected clutter data show good results when the smooth curve
chosen for curve fit is a third or fourth degree polynomia'.
The topics discussed briefly in this section require further
studies.
6.2.1 Clutter Model
The result of a preliminary investigation of the distri-
bution of clutter power received through a high PRF airborne
pulse doppler radar is presented in this section. The char-
acteristics of the radar used to collect the data and the
sample size are described in Table 6.2-1.
274
Table 6.2-1 Radar Characteristics and Clutter Sample Size
no. of pulses per range cell 175
no. of range cells per azimuth sample 40+
no. of azimuth samples 4
size of terrain covered
range 50-210 nmi
cross range 7.6 nmi at 210 nmi range
waveform high PRF
receiver bandwidth 1.25 MHz
carrier frequency S-band
Since the radar employs a high PRF wavaeform, the clutter
returns from over 200 nautical mile range swath are folded
into one approximately 3 nautical mile range interval which
is divided into some 40 or so range cells on the average. In
addition, these returns are modulated by the antenna beam
shape and the two way range attenuation on the one hand, and
by the antenna scanning and the platform motion on the
other. The platform is assumed to move at a speed of 360
knots at the altitude of 30,000 feet. Of the range cells
available, only samples from every other range cell are used
for analysis.
The objective of the investigation is to determine how
275
well the actual received clutter sample can be fit to the
nonstationary clutter model. The conventional view giving
rise to Weibull or lognormal distribution for clutter is
based on representing the clutter process as a wide sense
stationary random process. An alternate view, which is the
nonstationary characterization adopted in this investigation,
is modeling the clutter process as a time varying process
whose parameters are stationary within a radar detection de-
cision interval within a beam dwell.
This nonstationary characterization is described math-
ematically as follows: Let the instantaneous scattered clut-
ter power, which is proportional to the mean radar cross
section of the clutter patch or the clutter reflectivity, be
denoted by a random variable z, and the local mean of z be
denoted by another random variable u. Let p,(zlu) and P2 (u)
denote the conditional density of z given u and the prob-
ability density of u, respectively. The probability density
pl( ) corresponds to the short term or a local density for a
scattered clutter power while p2 ( ) corresponds to the
variation of u over time or space (entire terrain in the
surveillance volume).
The unconditioned density of z is given by
p(z) = fP l (zlu)P2 (u)du. (6.2-1)
276
Lewinski (24] has proposed gamma densities for the density
functions of both z and u which are expressed as
(k) kzk-l exp (-kz/u)pl(ZtIU) = r k(6.2-2)r(k) (u) k
and
(m)mum-1 exp(-mu/u)P 2 (u) - P(m) (U)m (6.2-3)
where overbar signifies the expected value, ( ) is the
gamma function, and k and m are the inverse of the normal-
ized variance of z and u (the square of the mean divided by
the variance), respectively. The parameter k is also known
as the shape factor while u is known as the scale factor.
Substituting the above expressions into Eqn. (6.2-1) yields
2(ki) (k+m)/2 (z) (k+m-2)/2p(z) = f(k)P(m) (u) (k+m)/2 Km62)
where Kp(.) is the modified Bessel function of second kind
of order p. When k = 1, p(z) becomes the density of the
K-distribution and when k = 1 and m = 1/2, p(z) becomes a
Weibull density with Weibull parameter equal to 1/2 [24).
The gamma density encompasses a large class of density
functions. When k = 1, it represent the exponential density
used in the Swerling case 1 and 2 models (see Section 2.6).
277
When k > 1, it represents the Rician density [29]. Actually,
it is slightly different from the Rician density commonly
used in the literature [10, 21, 23] for representing the re-
flected clutter power which is given as
p(z) = (l+a)/u] exp[-a-(l+a)z/u] Io(2[a(l+a)z/u]l/2)(6.2-5)
where a is the DC to AC power ratio and Io(. ) is the
modified Bessel function of the first kind of order zero.
The random variable u, the mean of z, is best obtained by
taking the mean of the sample values taken from multiple
carrier frequencies with the frequency spacing equal to or
larger than the waveform bandwidth. Since a single frequency
is used in the sample at hand, the average value of 175
pulses for each range cell is taken as the mean for that
range cell. It is assumed that a sufficient platform motion
and antenna scan occurred during the duration of the pulse
group to give a representative estimate of the mean.
The approximate density function for P2 (u) is obtained by
finding the value of m that gives a good fit between the
plot of Eqn. (6.2-3) (after multiplication by a number equal
to the area under the histogram) and the histogram of the
mean of each range cell in the test set. The approximate
density function for p(z) is similarly obtained by finding
the value of k that gives a good fit between the plot of
278
Eqn. (6.2-4) and the histogram of individual pulse returns
from all range-azimuth samples in the test set treated as a
single population. Note that knowing the value of k also
gives the conditional probability density function pl(zlu).
It is convenient for plotting purposes to use the normal-
ized gamma density according to the variable transformation,
u * u/u = y. Then, p(y) is given by
mm ym-i exp(-my)p(y) = - . (6.2-6)r(m)
The histogram of the normalized mean value of the clutter,
u/5, is shown in figure 6.2-1. The plots of Eqn. (6.2-6)
versus y = u/i for m = 1, and 2 are shown in Figure 6.2-2.
It can be seen that the gamma density with m = 2 is a good
fit to the histogram although the computed m is close to 1.
The histogram of individual pulse returns from all
range-azimuth samples in the test set, when matched to the
plot of a scaled version of Eqn. (6.2-4) represeenting p(z)
would allow the determination of k and pl(z/u). Unfortu-
nately, a way of plotting Eqn. (6.2-4) for a non-integer or-
der of the modified Bessel function of the second kind was
not found within the time constraint of this investigation
and the k value needed for pl(zlu) was not established by
this means.
An alternate approach was taken based on the assumption
279
41)0)
5.44)41)
C.,
0
%0
00
III~i 00 41
280
MULTIPUER-16 (AREA UNDER HSTOGRAd)2019
is1716
15
5 14A 13 i
-5 12
10-
I 6-7-
4
3
2
0
0 1 2 34 5
NORMALIM =n"rE PoSE VAL
(a) GAM MA DENSITY (SHAPE FACTOR(K)=1.0)
UULIPUE17- rE (AWA UMNO tISTOGRA )2019Io-
1716
15
5 14
t 13-5 12
9
7-
4 -
321
0 mom
0 2 3 4 5
NORMAL. D CLUTTER POWER ( Uzf)
(b) GAMMA DENSITY (SHAPE FACTOR(K)=2.0)
Figure 6.2-2Scaled Gamma Density Function with Shape Factor of 1 and 2
281
that the distribution of returns in each range cell all be-
long to a gamma density. The distribution of clutter returns
in individual range cells should reflect the conditional
density. Histograms of the normalized values of clutter re-
turns for a few range cells are shown in Figures 6.2-3 (a)
through (d). It is obvious that additional samples are
needed to give them a recognizable form. With the assump-
tion that returns in all range cells belong to a family of
gamma density, it can be conjectured that the histogram of
normalized clutter returns for a single range cell can be
approximated by averaging the same from all range cells.
The normalization is accomplished by dividing the variable
by its local mean (i.e., z/u) for each range cell. This is
done and the result is plotted in Figure 6.2-4. The result
reasonably matches with a Rician density of DC/AC ratio
equal to 40 which is plotted in Figure 6.2-5. The gamma den-
sity with the shape factor k equal to that computed from the
samples used (k - 27.5) is also plotted in Figure 6.2-5.
Based on visual inspection, it can be seen that a better
match is possible if a shape factor somewhat less than that
computed is used.
The most significant findings are that the mean and vari-
ance of the clutter returns vary significantly from one
range cell to the next. This contrasts with the frequently
used conventional assumption. The values for samples from
one azimuth look are listed in Table 6.2-2 for illustration.
282
SAULE SIZE-I 75. IM.AN-,7sT7Z.
40-
50-
20-
10-
0 0.2 0.4 0.6 0O8 1 1.2 1.4 1.6 is8 2
NORW4.JZED CLL11f POWER(/,
(a)- HISTOGPAM OF CLUTTER RETURN (R-c-
SMLE SrlE-175. MEM.-I363p,.70--
so.
40-
30
20-
10-
0-,o
0 0.2 0.4 0.6 0.6 1 1.2 1.4 1.6 1.A 2
NOftMAUZED cLUlER Po (/.
(b) HISTOGRAM OF CLUTTER RETURN~e4 2o7
Figure 6.2-3Histograms of Normalized Clutter Returns
for Individual Range cells
283-
SAUIX qSJE-175. MtAN-Z7M
70
40
80
E
20-
to
10
0 0.2 0.4 0.4 0. 1 1.2 1.4 1.6 is 2
NORMXM~ 0±L1E POW~ (ANI
(c) HISTOGRAM OF CLUTTER RETURN ('4-4c)
SMXLE 9E-l 75& W~AN-%70
50-aO
30
to- fl-
0-
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1. 1.8 2
N#40UUZD CUTR POW (?AL)
(d) HISTOGRAM OF CLUTTER RETURN (4-,7)
Figure 6.2-3
Histograms of Normalized Clutter Returnsfor Individual Range Cells (cont'd)
284
-~~~~~' -- - - - -
0
- -Iq
-i - -
Azpanwb
285'~
2 ULnTIPI-7 (A/RCA LN0DL rR E iSOGRAU)2019
10
1716-
§ I.is14
13-
.J 122 11
7
4
32
0
a 0.2 04 0.6 0.8 1 1.2 1.4 I. 1.6 2
(a) RICIAN DENSITY (DC/AC=40)
20-MULUWPI-7. (.'RV Wi" 1W IS, NSRAU)20-
19-
1-17
14
S 12
11I'
N 10
.7I!
.32-
o 0.2 0.4 0.4 0.6 1 1.2 1.4 1.6 1.8 2
(b) GAMMA DENSITY (SHAPE FACTOR(K)=27.5)
Figure 6.2-5Candidate Rician and Gamma density
for the Conditional Density for the Clutter
286
Table 6.2-2
Range Cell to Range Cell Variation of Clutter Statistics
clutter power statistics
range cell # mean variance max. value min. value
7 136,383 2.2xi0 0 9 254,530 86,725
9 2,528,749 5.8xi01 1 3,257,461 939,497
11 464,083 3.5xi01 0 914,689 197,640
15 382,682 6.9xi00 9 517,985 153,493
17 90,185 1.7x100 9 141,520 13,253
19 1,706,289 1.6x101 0 1,946,404 1,566,850
21 338,809 1.3x101 0 472,066 122,509
23 1,225,326 7.7xi00 9 1,376,500 1,116,757
25 1,677,916 9.0x101 0 2,002,568 1,081,825
27 229,145 4.0x100 9 322,592 117,625
29 581,995 2.8xi01 0 941,845 411,956
33 458,439 2.2x101 0 862,948 332,100
35 2,431,334 1.8x101 1 3,075,201 1,667,818
37 3,427,057 6.2xi01 0 3,686,800 2,877,200
39 668,303 5.1x101 0 1,118,617 388,145
43 751,256 5.8xi0 I0 1,039,725 273,320
45 277,727 7.2x10 0 8 327,625 242,045
47 166,344 2.2xi00 9 238,954 113,074
287
6.2.2 Clutter Cancellation Techniques
In this section a performance comparison of three types
of clutter cancellers is made using live data collected from
a high PRF airborne pulse doppler radar. The three types of
clutter cancellers compared are
a. Velocity shaped IIR filter.
b. Delay line canceller with binomial coefficients as
its weight.
c. Optimum linear filter whose weights are derived from
the eigen vector corresponding to the minimum eigen
value of the clutter correlation matrix.
The velocity shaped IIR filter consists of two sections
of 2-pole, 2-zero recursive digital filters in cascade whose
coefficients are chosen so as to give a flat magnitude re-
sponse above its cutoff frequency which can be selected for
either 70 or 90 knots.
The filters of (b) and (c) above are known also as trans-
versal filters whose outputs are a weighted sum on a sliding
window of the input sequence or the input vector. The out-
put sequences of these filters are weighted and summed again
to produce a specific doppler filter output. A representa-
tion of the above processes in terms of matrix multiplica-
tions is given by Andrews (30].
288
According to linear optimum filter theory, the above pro-
cess can be performed by one operation. This is expressed
mathematically as
x = W*Y (6.2-7)
where x is the scalar output, Y = Z+S is the input column
vector consisting of the clutter plus thermal noise inter-
ference vector Z and the signal vector S, and W* is the
transpose of the complex conjugate of the weight vector W
defined as
w M M -is (6.2-8)
and M is the correlation matrix of the interference process
assumed to be zero mean [31].
One difficulty with this approach is that the signal vec-
tor is a priori unknown. In this case the signal vector, in
particular its doppler frequency, is assumed to be equally
likely to be anywhere within frequency interval correspond-
ing to one PRF. Then, the optimum weight is given by the
eigen vector corresponding to the minimum eigen value of the
interference correlation matrix (32, 33].
The real difficulty with this optimum filter is that the
interference correlation matrix is also unknown and cannot
289
be approximated from the adjacent range gate samples as as-
sumed by others [31, 34, 35]. This is because the clutter
statistics are different from one range cell to the next as
shown in the previous sectione Furthermore, in a high PRF
radar the dimension of the correlation matrix is large
typically on the order of 128 or larger; and the operation
indicated by Eqn. (6.2-7) and (6.2-8) is extremely diffi-
cult, if not impossible, to carry out. While M is a positive
definite Hermitian matrix in theory, an examination of real
clutter data shows that it-is highly ill conditioned. The
difficulty arises due to the high degree of clutter correla-
tion from pulse to pulse. At the same time, the correlation
is not perfect enough with the result, when binomial coef-
ficients are used as the weight, that the performance of the
delay line canceller deteriorates rather than improves as
the number of canceller stages increases beyond two. In ad-
dition Hsiao [37] shows, even under ideal assumptions such
as a known exponential auto-correlation function for the
clutter, that the increase in the clutter canceller improve-
ment factor diminishes rapidly as the number of stages is
increased beyond 7 or 8. These considerations limited the
number of stages for transversal filters examined in the
performance comparison to 7 making it an 8 pulse canceller
at the maximum.
In the data sample examined, there are approximately 175
pulses available for processing after the multiple time
290
around echoes settle down in each CPI (see Figure 2.1-2).
With the velocity shaped IIR filter, a 128-point FFT follows
the clutter canceller after those pulses in the early por-
tion of the filter output are discarded during which the
filter goes through a transient period (see Figure 2.1-3).
For the transversal filters, there are no transient periods.
Thus, (175-N) output pulses are zero padded for a 256-point
FFT where N is the number of canceller stages.
The frequency span equal to one PRF of the data set cor-
responds approximately to a velocity span of 0 to 2,454
knots. In each trial, a synthetic target is injected with
varying amplitude at a doppler frequency corresponding to a
specific target velocity. The results with the velocity
shaped IIR filter are shown in Figures 6.2-6 (a) through
(d) together with those for a 2-stage delay line canceller
with binomial coefficients as its weights for comparison.
The two humps appearing near the right edge in the figure
correspond to (from left to right) the altitudeline clutter
and the first sideobe clutter, repectively. Different adap-
tive constant false alarm rate (CFAR) thresholds are used in
these range-doppler zones in the actual radar implementa-
tion. The results obtained using delay line cancellers with
binomial coefficient weights for canceller stages of 2, 3,
and 4 are shown in Figure 6.2-7. The canceller transfer
function (gain as a function of frequency) is superimposed
on the figure for visual aid. Increasing the number of
291
__ _ -rccU t~ I__ ~H i l
S.4
31
zk I~2
C - ~ U M41
-l on a 0
0 I I0 aE H~ ~ _
llo -,
II0. F 0~ X
4A.
-00
1--96
crZ
292
-- III E *NitIme
CL IZE
cc, o- o- a a =n - -
04
0 -P
At! :c -P4 r
MA1 04 u
S C
96 ZI...
.rC ON
=66' EP
.0 04 0aa a
ZGD. l~V M'm0fl~ (Df) 3aaflfl Lwz~s~od ZUrl
293
CLa 6
I 03
Le I I - - ga
we1 we
U, L lI I, WtI V
iU C4
()3OLM(gp) 3GD.io'vx A c~
4
{1k- pI V4 ~r
1H'L~a~ IIo-
*4 0
294
stages beyond two resulted in performance degradation. This
is due to imperfect pulse-to-pulse clutter correlation. A
comparison of 2-stage and 7-stage cancellers is shown in
Figure 6.2-8. The performance deterioration is even more
evident.
The above results are for samples from range cell 33 at a
particular azimuthal direction. The results for different
range cells in the same azimuth set with the eigen vector
obtained from the estimated clutter correlation matrix as
filter weight are shown in Figures 6.2-9. Also shown in the
figure in parallel is the corresponding results with the IIR
filter. The transfer function for the transversal filter is
shown in Figure 6.2-10 for each of the range cells examined.
In computing the eigen vector, a sample correlation matrix,
which is an average of a set of 8x8 correlation matrices
each formed by the outer product of 8 pulses taken at 16 or
32 pulse interval from the received pulse train for the test
cell, is first obtained. The eigen values and eigen vectors
for the sample correlation matrix for each range cell sample
are then computed using the EISPACK computer program devel-
oped by Argonne National Laboratory on a mini VAX computer.
Without exception, in all trials some eigen values turned
out to be negative. The eigen vector corresponding to the
minimum positive eigen value was used to provide filter
weights for each range cell sample.
295
Maqnitude vs. Frequency (Tgt Spd=1OO0cts)3P,096 DeCaW Ls CV~cmir
70
40-
synthetic target
40 amplitude=5 quanta
20
io
10-
Mtagnlitude- vsFeuncuTtap~Oks
50-
4 ythtctre
30 piud= u
230
229
(a).Magnitude vs Frequency (Tgt Spd=1O0kts)in Clakk Caiaoer
so-synthetic target
45- 17 sidelobe clutter
40- altitudeline clutter
35-
* 0-OU
Ila3IIIIIHII
5
40
36
30
60
46
Fiur 3.-9
269
(c) Magnitude v!5 Frequency (Tgt Spd= I OCkt3)R citOw CanM&Wa
synthetic target clutteramplitude-2 quanta
40 /
* 30
r
C2g
* 20
Io
o is 31 47 43 70 9 111 127
FPT RI~ Nwr&w
(d) Magnitude vs Frequency (Tgt Spd=1OOkts)Epww~cW 4 LeflM 6 Umd m Vi
synthetic targetamplitude-2 quanta clutter
40
3 0
20
- 10
-0
-0 Q
32 64 129 140 192 224 254
Mi v~Iw *.#qbwI
Figure 6.2-9
Performance Comparison of Clutter Canceller:IIR vs. FIR with Eigen Vector as Weights (cont'd)
298
(e) Magnitude vs Frequency (Tgt Spd=1OO -3}WR Ch~le cwceftr
50synthetic targetamplitude-2 quanta clutter
40
C20
00
0 ---- ---.... ..
0 s 3 l) 9 1
(f Mantd sFrqec Tt p=0ks
v 20
10
Peformagnce CompFrquencyf (Tgte Cadnceller)IIR vs.E~mv~ FI with Eie Vetora egt~ot
5299
(g) Magnituo-e vs Fretquency jgt -'pd=10c0t3IR Chm"e Cccv,
50-synthetic targetampl'tude-2 quanta
40-
20U
10
o is 3 40 64 so 96 12 128
(h) Magnitude vs Frequency (Tgt Spd= 1 OOkts)Vqrvcwo Lwe h S'&WasS4
50
synthetic targetamplitude-2 quanta
j20
- 10
o M2 64 96 201 160 192 :-A 25.3
Figure 6.2-9
Performance Comparison of Clutter Canceller:IIR vs. FIR with Eigen Vector as Weights (cont'd)
300
(1) Magnitude vSz Frequiency (Tgt Spa=!~ .tsMR Clutter Caurceller
synthetic targetam litude=2 quanta
40
In
S30
* 20
010
(iMagnitude vs Frequency (Tgt Spd=1O0kts)Ei~aWr of Laft~h 6 UW as Wi
s-synthetic targetamplitude-2 quanta
40
30C
20
U 0
30
%0
1% C9-4 H.
~-0 A0a
N 0-1 40 ~
r-4 01*1U 4>,
(gp) spn4Tu69H
302
These results can be compared by visual inspection (the
best knowledge based detector). While subjective, it is ap-
parent that no one technique is noticeably superior to the
other for discriminating the injected synthetic target from
the clutter. As a matter of fact, performance of the opti-
mum filter (filter with the eigen vector as its weight)
seems inferior to that of conventional filters used almost
exclusively in radar systems operating today. However, a
higher degree of reduction in amplitude of the altitudeline
and the first sidelobe clutter away from the mainbeam clut-
ter is quite evident when the optimum filter is employed.
303
Bibliography
1. E. Brookner, Radar Technology, Artech House, Inc. 1977.
2. DiFranco and Rubin, Radar detection, Artech House, Inc.
1980.
3. M. Skolnik, Radar Handbook, McGraw-Hill Company, 1970.
4. D. K. Barton, Radar System Analysis, Artech House, Inc.
1976.
5. A. Popoulis, Probability, Random Variables and Stochas-
tic Processes, McGraw-Hill Company, 1965.
6. J. D. Mallet and L. E. Brennan, "Cumulative Prob-
ability of Detection for Targets Approaching a Uni-
formly Scanning Search Radar'" Proceedings of the IEEE,
vol.,51, No. 4, pp. 596-601, April, 1963.
7. L. Blake, "The Effective Number of Pulses per Beamwidth
for a Scanning Radar," Proc. IRE, 41, No. 6 pp. 770-774,
June, 1953.
8. M. Schwartz, "A Coincidence Procedure for Signal Detec-
tion," Trans. IRE, vol. IT-2, No. 4, pp. 135-139, Dec.
1956.
9. M. Skolnik, Introduction to Radar Systems, McGrow Hill
Book Company, New York, 1962.
10. F. E. Nathanson, Radar Design Principles, McGraw Hill
Book Co., New York, 1969.
11. V. C. Vannicola and J. A. Mineo, "Application of Knowl-
edge Based Systems to Surveillance," 88 CH
2572-6/88/0000-0157 $01.00, 1988 IEEE.
304
*12. A. Corbeil, et al., Track Before Detect Development and
Demonstration Program, Phase II, RADC-TR-86-68, May,
1986.
13. I. S. Reed and R. M. Gargliardi, "A Recursive Moving-
Target-Indication Algorithm for Optimum Image
Sequences," IEEE Trans. on Aerospace and Electronic Sys-
tems, Vol. 26, NO. 3, May 1990.
14. S. Haykin, Communication Systems, 2nd edition, John
Wiley & Sons, New York, 1983.
15. P. Swerling, "Probability of Detection for Fluctuating
Targets, Special Monograph," Trans. IRE on Information
Theory, IT-6: (2). PP269-308, April, 1960.
16. R. R. Booth, "The Weibull Distribution Applied to the
Ground Clutter Backscatter Coefficient," U.S. Army Mis-
sile command, Redstone Arsenal, AL, Report RE-TR-69-15,
June, 1969.
17. R. L. Fante, "Detection of Multiscatter Targets in
K-Distributed Clutter," IEEE Trans. on Antennas and
Propagations, AP-32, 12, pp. 1358-1363, 1984.
18. A. Farina, A. Russo, and F. Scannapieco, "Radar Detec-
tion of Target Signals in Non-Gaussian Clutter: theory
and Applications," Chinese Intl. Radar Conference,
Nnanjing, pp. 442-449, November 1986.
19. G. Fedele, L. Izzo, and L. Paura, "Optimum and
Suboptimum Space Diversity Detection of Weak Siganls in
Non-Gaussian Noise," IEEE Trans. Comm., vol. COM-32, pp.
990-997, Sept. 1984.*Although this report references limited document (item 12), no limitedinformation has been extracted. Dist Statement - DOD & DOD contractorsonly; critical technology; May 86.
305
20. Conte, Izzo, Longo, and Paura, "Asymtotically Optimum
Radar Detection in Sea Clutter," MELECON 85, Madrid,
Oct. 1985.
21. TD, edited by D. C. Schleher, Artech House, 1980.
22. Optimized RADAR PROCESSORS, edited by A. Farina, IEE RA-
DAR, SONAR, NAVIGATION and AVIONICS Series 1, Peter
Peregrinus Ltd., 1987.
23. FAA-E-2763, Federal Aviation Administration Specifica-
tion, "ARSR-4 RADAR SYSTEM. APPENDIX A: Radar Clutter
Model," U.S. Dept. of transportation, Oct. 1986.
24. D. J. Lewinski, "Nonstationary Probabilitic Target and
Clutter Scattering Models," IEEE Trans., vol. AP-31, No.
3, Kay, 1983.
25. S. Watts, "Radar Detection Prediction in K-Distributed
Sea Clutter and Thermal Noise," IEEE Trans., vol AES-23,
No. 1, pp. 40-45, Jan. 1987.
26. E. Jakeman and P. Pusey, "A Model for Non-Rayleigh Sea
Echo," IEEE Trans., vol. AP-24, PP 806-814, Nov. 1976.
27. R. J. Miller, "A Coherent Model of Radar Clutter," Proc.
of the 1988 IEEE National Radar Conference, pp. 246-248.
28. Greenstein, Brindley, and Carlson, "A Comprehensive
Ground Clutter Model for Airborne radars," IIT Research
Institute, Sept. 1969.
29. M. Nakagami, "The M Distribution -A General Formula of
Intensity Distribution of Rapid Fading," in Statistical
Methods in Radio Wave ProDaiation, W. C. Hoffman, Ed.,
Pergamon, NY 1960.
306
30. G. A. Andrews, Jr., "Performance of Cascaded MTI and Co-
herent Integration Filters in a Clutter Environment,"
Naval Research Lab., Washington, D.C., March 27, 1973
31. I. Reed, J. Mallett, and L. Brennan, "Rapid Convergence
in Adaptive Arrays," IEEE Trans., vol. AES-10, No. 6,
pp. 853-863, Nov. 1974.
32. R. C. Emerson, "Some Pulsed Doppler MTI and MTI
Tehniques," Rand Corp., Report R-274, pp. 1-124, March,
1954.
33. B. Noble and J. Daniel, ADolied Linear AlQebra,
Prentice-Hall, Inc. New Jersey, 1977.
34. E. J. Kelly, "Adaptive Detection in Nonstationary Inter-
ference, Part I and Part II," MIT Lincln Lab. Tech. Re-
port 724, June, 1985.
35. E. D'Addio, A. Farina, and E. Studer, "Performance Com-
parison of Optimum and Conventional MTI and Doppler Pro-
cessors," IEEE Trans. on Aerospace and Electronic Sys-
tems, Vol. AES-20, NO. 6, pp. 707-715, November 1984.
36. J. K. Hsiao, "On The Optimization of Clutter Rejection,"
IEEE Trans., vol. AES-10, No. 5, pp. 622-629, Sept.
1974.
307
MISSION
OF
ROME LABORATORY
Rome Laboratory plans and executes an interdisciplinary program in re-search, development, test, and technology transition in support of Air
Force Command, Control, Communications and Intelligence (C 31) activitiesfor all Air Force platforms. It also executes selected acquisition programsin several areas of expertise. Technical and engineering support withinareas of competence is provided to ESD Program Offices (POs) and otherESD elements to perform effective acquisition of C31 systems. In addition,Rome Laboratory's technology supports other AFSC Product Divisions, theAir Force user community, and other DOD and non-DOD agencies. RomeLaboratory maintains technical competence and research programs in areasincluding, but not limited to, communications, command and control, battlemanagement, intelligence information processing, computational sciencesand software producibility, wide area surveillance/sensors, signal proces-sing, solid state sciences, photonics, electromagnetic technology, super-conductivity, and electronic reliability/maintainability and testability.